Search

For a compact set A \subset \mathbb {R}^d and an integer k\ge 1, let us denote by \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} the Minkowski sum of k copies of A. A theorem of Shapley, Folkmann and Starr (1969) states that \frac {1}{k}A[k] converges to the convex hull of A in Hausdorff distance as k tends to infinity. Bobkov, Madiman and Wang [ Concentration, functional inequalities and isoperimetry , Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of \frac {1}{k}A[k] is nondecreasing in k, or in other words, in terms of the volume deficit between the convex hull of A and \frac {1}{k}A[k], this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if d=1 but fails for any d \geq 12. In this paper we show that the conjecture is true for any star-shaped set A \subset \mathbb {R}^d for d=2 and d=3 and also for arbitrary dimensions d \ge 4 under the condition k \ge (d-1)(d-2). In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in \mathbb {R}^d, for any d \geq 7. [ABSTRACT FROM AUTHOR]

In this paper we extend the calculation of the geometric Waldspurger periods from our paper [Compos. Math. 144 (2008), no. 2, 377-438] to the case of ramified coverings. We give some applications to the study of Whittaker coeffcients of the theta-lifting of automorphic sheaves from PGL2 to the metaplectic group SL2; they agree with our conjectures from [Geometric Whittaker models and Eisenstein series for Mp2, arXiv:1221.1596]. In the process of the proof, we construct some new automorphic sheaves for GL2 in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair (SL2, PGL2). [ABSTRACT FROM AUTHOR]

The g -extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464–473] determined the 3 -extra edge-connectivity of balanced hypercubes B H n and conjectured that the g -extra edge-connectivity of B H n is λ g ( B H n) = 2 (g + 1) n − 4 g + 4 for 2 ≤ g ≤ 2 n − 1. In this paper, we confirm their conjecture for n ≥ 6 − 1 2 g + 1 and 2 ≤ g ≤ 8 , and disprove their conjecture for n ≥ 3 e g ( B H n) g + 1 and 9 ≤ g ≤ 2 n − 1 , where e g ( B H n) = max { | E ( B H n [ U ]) | | U ⊆ V (B H n) , | U | = g + 1 }. [ABSTRACT FROM AUTHOR]

Let (G) and e(G) denote the domination number and exponential domination number of graph G, respectively. Henning et al., in [Hered-itary equality of domination and exponential domination, Discuss. Math. Graph Theory 38 (2018) 275-285] gave a conjecture: There is a finite set F of graphs such that a graph G satisfies (H) = e(H) for every induced subgraph H of G if and only if G is F-free. In this paper, we study the conjecture for subcubic graphs. We characterize the class F by minimal forbidden induced subgraphs and prove that the conjecture holds for subcubic graphs. [ABSTRACT FROM AUTHOR]

A graph G is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum as G. Let k, ti (1 ≤ i ≤ k), and s be natural numbers. A path-friendship graph, Gs,t1...,tk, is a graph of order n = 2s + t1 + ... + tk+1 which consists of s triangles and k paths of lengths t1,t2 ..., tk sharing a common vertex. In this paper, we show that these graphs are DQS and using this result, we respond to a conjecture in [F. Wen, Q. Huang, X. Huang and F. Liu, The spectral characterization of wind-wheel graphs, Indian J. Pure Appl. Math. [ABSTRACT FROM AUTHOR]

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 ∈ T, and G be a graph. In the paper we study relations between the T-edge spans espT (G) and espd⊙T (G), where d is a positive integer and d ⊙ T = {0 = t = d (max T + 1): d | t ⊙ t/d ⊙ T}. We show that espd⊙T (G) = d espT (G) - r, where r, 0 = r = d - 1, is an integer that depends on T and G. Next we focus on the case T = {0} and show that espd⊙T{0}(G) = ⊙d(c(G) - 1), where c(G) is the circular chromatic number of G. This result allows us to formulate several interesting conclusions that include a new formula for the circular chromatic number c(G) = 1 + inf espd⊙{0}(G)/d: d = 1 and a proof that the formula for the T-edge span of powers of cycles, stated as conjecture in [Y. Zhao, W. He and R. Cao, The edge span of T-coloring on graph Cd n, Appl. Math. Lett. 19 (2006) 647-651], is true. [ABSTRACT FROM AUTHOR]

Abstract Let R be a noncommutative ring with unity. The commuting graph of R , denoted by Γ (R) , is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if x y = y x. Let F be a finite field and n ≥ 2. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if Γ (R) ≅ Γ (M n (F)) , then R ≅ M n (F). In this paper, we prove the conjecture whenever n is of the form 2 k 3 l with k ≠ 0. [ABSTRACT FROM AUTHOR]

Guide rails’ nonlinear dynamic response to the collision process between gangues and the suspended buffer is the basis for its intensity examination and fatigue prediction. This paper established a numerical model for simulating the dynamic response of buffer’s guide rails using Pro/E and Femap based on an analysis of the basic structural components of the suspended buffer. The effects of the gangues diameter and feeding rate on the guide rails’ dynamic response were investigated, which were mainly revealed by a tension in guide rails. Simulation results showed that the tension in guide rails rapidly increased to a peak value after the suspended buffer was impacted by gangues, followed by a periodic vibration. The peak tension, maximum and minimum tensions within the periodic vibration were exponential functions of the gangues diameter. And they depended linearly on the feeding rate. The vibration frequency of the tension in guide rails was an exponential function of the gangues diameter but did not depend on the feeding rate. Based on the backfilling workface parameters used in D.Ping coal miner’s 15601 working panel, the feeding rate of 500 t/h was selected. The diameter of the crushed gangues was selected to be 50 mm based on the strength of guide rails. Their expected service life is 54.2 years. Finally, an industrial test was performed in D.Ping coal miner’s 15601 working panel and the guide rails operated steadily. The measured vibration parameters of the tension in guide rails agree well with that of numerical predictions, which verified the reliability of the numerical model to some extent. [ABSTRACT FROM AUTHOR]

Although the deterministic equation of life predicted and recognized by Albert Einstein is not yet defined, there are clear marks and signs of mathematical regularity which appear in nature showing fascinating accuracy. One of those "God's imprints" in the world that surrounds us is the Fibonacci Sequence, which represents more likely a starting point in revealing certain universal formula of life. The main idea of this paper is to enlighten two points: the Fibonacci sequence and the Golden Ratio, as well as to stress out the importance of reveling the perfect mathematical patterns that exist both in the organic as well as in the inorganic world. Thus, we might be able to understand the physical, spiritual and mental aspect of a human nature more clearly and completely. [ABSTRACT FROM AUTHOR]

This paper describes a web server developed for designing therapeutic peptides with desired half-life in blood. In this study, we used 163 natural and 98 modified peptides whose half-life has been determined experimentally in mammalian blood, for developing in silico models. Firstly, models have been developed on 261 peptides containing natural and modified residues, using different chemical descriptors. The best model using 43 PaDEL descriptors got a maximum correlation of 0.692 between the predicted and the actual half-life peptides. Secondly, models were developed on 163 natural peptides using amino acid composition feature of peptides and achieved a maximum correlation of 0.643. Thirdly, models were developed on 163 natural peptides using chemical descriptors and attained a maximum correlation of 0.743 using 45 selected PaDEL descriptors. In order to assist researchers in the prediction and designing of half-life of peptides, the models developed have been integrated into PlifePred web server (). [ABSTRACT FROM AUTHOR]

The atom-bond connectivity index (or, for short, ABC index) is a molecular structure descriptor bridging chemistry to graph theory. It is probably the most studied topological index among all numerical parameters of a graph that characterize its topology. For a given graph G = (V, E), the ABC index of G is defined as , where di denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j. A combination of physicochemical and the ABC index properties are commonly used to foresee the bioactivity of different chemical composites. Additionally, the applicability of the ABC index in chemical thermodynamics and other areas of chemistry, such as in dendrimer nanostars, benzenoid systems, fluoranthene congeners, and phenylenes is well studied in the literature. While finding of the graphs with the greatest ABC-value is a straightforward assignment, the characterization of the tree(s) with minimal ABC index is a problem largely open and has recently given rise to numerous studies and conjectures. A B1-branch of a graph is a pendent path of order 2. In this paper, we provide an important step forward to the full characterization of these minimal trees. Namely, we show that a minimal-ABC tree contains neither 4 nor 3 B1-branches. The case when the number of B1-branches is 2 is also considered. [ABSTRACT FROM AUTHOR]

The leaf distance of a tree is the maximum d such that the distance between any pair of leaves in the tree is at least d . Kaneko provided sufficient conditions to force the existence of a spanning tree with leaf distance at least d = 3 and conjectured that similar conditions suffice for larger d . The case when d = 4 was later proved by Kaneko, Kano, and Suzuki. In this paper, we show that when d ≥ 4 , a stronger form of this conjecture holds for graphs with independence number at most five. As an immediate corollary, we obtain that when d ≥ n ∕ 3 , this stronger version holds for all n -vertex graphs, consequently proving Kaneko’s conjecture for d ≥ n ∕ 3 . [ABSTRACT FROM AUTHOR]

Using the spectrum data for quality prediction always suffers from noise and colinearity, so variable selection method plays an important role to deal with spectrum data. An efficient elastic net with regression coefficients method (Enet-BETA) is proposed to select the significant variables of the spectrum data in this paper. The proposed Enet-BETA method can not only select important variables to make the quality easy to interpret, but also can improve the stability and feasibility of the built model. Enet-BETA method is not prone to overfitting because of the reduction of redundant variables realized by elastic net method. Hypothesis testing is used to further simplify the model and provide a better insight into the nature of process. The experimental results prove that the proposed Enet-BETA method outperforms the other methods in terms of prediction performance and model interpretation. [ABSTRACT FROM AUTHOR]

This paper estimates regional tornado risk from records of past events using statistical models. First, a spatial model is fit to the tornado counts aggregated in counties with terms that control for changes in observational practices over time. Results provide a long-term view of risk that delineates the main tornado corridors in the United States where the expected annual rate exceeds two tornadoes per 10,000 square km. A few counties in the Texas Panhandle and central Kansas have annual rates that exceed four tornadoes per 10,000 square km. Refitting the model after removing the least damaging tornadoes from the data (EF0) produces a similar map but with the greatest tornado risk shifted south and eastward. Second, a space-time model is fit to the counts aggregated in raster cells with terms that control for changes in climate factors. Results provide a short-term view of risk. The short-term view identifies a shift of tornado activity away from the Ohio Valley under El Niño conditions and away from the Southeast under positive North Atlantic oscillation conditions. The combined predictor effects on the local rates is quantified by fitting the model after leaving out the year to be predicted from the data. The models provide state-of-the-art views of tornado risk that can be used by government agencies, the insurance industry, and the general public. [ABSTRACT FROM AUTHOR]

Spatial-temporal k-anonymity has become a mainstream approach among techniques for protection of users’ privacy in location-based services (LBS) applications, and has been applied to several variants such as LBS snapshot queries and continuous queries. Analyzing large-scale spatial-temporal anonymity sets may benefit several LBS applications. In this paper, we propose two location prediction methods based on transition probability matrices constructing from sequential rules for spatial-temporal k-anonymity dataset. First, we define single-step sequential rules mined from sequential spatial-temporal k-anonymity datasets generated from continuous LBS queries for multiple users. We then construct transition probability matrices from mined single-step sequential rules, and normalize the transition probabilities in the transition matrices. Next, we regard a mobility model for an LBS requester as a stationary stochastic process and compute the n-step transition probability matrices by raising the normalized transition probability matrices to the power n. Furthermore, we propose two location prediction methods: rough prediction and accurate prediction. The former achieves the probabilities of arriving at target locations along simple paths those include only current locations, target locations and transition steps. By iteratively combining the probabilities for simple paths with n steps and the probabilities for detailed paths with n-1 steps, the latter method calculates transition probabilities for detailed paths with n steps from current locations to target locations. Finally, we conduct extensive experiments, and correctness and flexibility of our proposed algorithm have been verified. [ABSTRACT FROM AUTHOR]

A complex projective manifold is said to be weakly special if none of its finite étale covers has a dominant rational map to a positive-dimensional manifold of general type. Weakly special manifolds are conjectured by Harris and Tschinkel (Rational points on quartics, Duke Math. J. 104 (2000), 477–500) to be potentially dense if defined over a number field. In a previous paper by F. Campana (Special varieties, orbifolds and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–665), the stronger notion of special (complex projective) manifolds was introduced. These are conjectured in (9.20) of that work to be exactly the potentially dense manifolds (when defined over a number field). The two notions of specialness and weak-specialness coincide up to dimension 2, but differ from dimension 3 on, as shown by examples $X$ constructed by Bogomolov and Tschinkel (Special elliptic fibrations, in Proc. Fanu Conf., Torino, 2003, ed. A. Conte), answering a question raised in Campana''s paper. These examples should thus be potentially dense, according to the first conjecture, but should not be so, according to the second conjecture. The present techniques of arithmetic geometry do not seem to be able to decide between these two conflicting conjectures. So, instead of this, we investigate the more tractable hyperbolic aspects of the examples of Bogomolov and Tschinkel, and show that for certain manifolds $X$, the behaviour of entire transcendental curves is that given in Conjecture 9.16, p. 618 of Campana''s above-mentioned work. More precisely, the image of any holomorphic map from ${\mathbb{C}}$ to $X$ is contained either in some fibre of the unique elliptic fibration on $X$, or in some fixed divisor of $X$. This is consistent with the standard links (Lang and Vojta for varieties of general type, the first author''s previous work in general) expected between the hyperbolic and arithmetic behaviour of projective varieties in cases where the second conjecture (stated by Campana) holds. Conversely, if the conjecture stated by Harris and Tschinkel is true, these expected links between arithmetics and complex hyperbolicity should fail to hold on these examples already. [ABSTRACT FROM AUTHOR]

In this paper we will first give a positive answer to Kaiser's conjecture on ε -positive centers for convex curves and then present its two applications. [ABSTRACT FROM AUTHOR]

The performance of objective speech and audio quality measures for the prediction of the perceived quality of frequency-compressed speech in hearing aids is investigated in this paper. A number of existing quality measures have been applied to speech signals processed by a hearing aid, which compresses speech spectra along frequency in order to make information contained in higher frequencies audible for listeners with severe high-frequency hearing loss. Quality measures were compared with subjective ratings obtained from normal hearing and hearing impaired children and adults in an earlier study. High correlations were achieved with quality measures computed by quality models that are based on the auditory model of Dau et al., namely, the measure PSM, computed by the quality model PEMO-Q; the measure qc, computed by the quality model proposed by Hansen and Kollmeier; and the linear subcomponent of the HASQI. For the prediction of quality ratings by hearing impaired listeners, extensions of some models incorporating hearing loss were implemented and shown to achieve improved prediction accuracy. Results indicate that these objective quality measures can potentially serve as tools for assisting in initial setting of frequency compression parameters. [ABSTRACT FROM AUTHOR]

Abstract: Wegner conjectured that for each planar graph with maximum degree at least 4, if , and if . Let be a planar graph without 4-cycles. In this paper, we discuss the -labelling of , and show that , where and are positive integers with . As a corollary, and . [Copyright &y& Elsevier]

In this paper we will algorithmically prove the global epsilon constant conjecture for all Galois extensions L/Q of degree at most 15. In fact, we will obtain a slightly more general result whose proof is based on an algorithmic proof of the local epsilon constant conjecture for Galois extensions E/Qp of small degree. To this end we will present an efficient algorithm for the computation of local fundamental classes and address several other problems arising in the algorithmic proof of the local conjecture. [ABSTRACT FROM AUTHOR]

The domination number of a graph G is the smallest number of vertices which dominate all remaining vertices by edges of G. The bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with domination number greater than the domination number of G. The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations. This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers. [ABSTRACT FROM AUTHOR]

For an edge uv of a graph G, the weight of the edge e = uv is defined by w(e) = 1/ √d(u)d(v). Then R(G)= ΣuvϵE(G) 1/√d(u)d(v)= ΣeϵE(G)w(e) is called the Randić index of G. If G is a connected graph, then rad(G)=minx maxy d(x,y) is called the radius of G, where d(x;y) is the distance between two vertices x;y. In 2000, Caporossi and Hansen conjectured that for all connected graphs except the even paths, R(G) ≥ r(G). They proved the conjecture holds for all trees except the even paths. In this paper, it is proved that the conjecture holds for all unicyclic graphs, bicyclic graphs and some class of chemical graphs. [ABSTRACT FROM AUTHOR]

Abstract: While some educators argue that teacher–student gender matching improves student performance, there is little empirical evidence to support this hypothesis. This paper assesses the impact of teacher–student gender matching on academic achievement across fifteen OECD countries using data from the Trends in International Mathematics and Science Study (TIMSS). One attractive feature of TIMSS is that it provides information on test scores and teacher characteristics, including gender, for both math and science thereby allowing for student fixed effects estimation. The results provide little support for the conjecture that students benefit from teacher–student gender matching. [Copyright &y& Elsevier]

The article discusses the key concerns in the main conjecture of equivariant Iwasawa theory. The Lifted Root Number suggests a refinement of the Main Conjecture of classical Iwasawa theory. The article is based on classical works done by Iwasawa, Kubota, Leopoldt, Greenberg, Ferrero Washington, and Mazur Wiles. The authors present theorems and proofs in this paper to further explain their findings.

Abstract: We investigate the functional code introduced by G. Lachaud (1996) in the special case where X is a non-singular Hermitian variety in and . In , F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) on the minimum distance of this code for a Hermitian variety X in . In this paper, we will answer the question about the minimum distance in general dimension N, with . We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes. [Copyright &y& Elsevier]

Abstract: In this paper we treat cyclotomic binary duadic codes. The conjecture of Ding and Pless is that there are infinitely many cyclotomic duadic codes of prime lengths that are not quadratic residue codes. We shall prove this conjecture by using the special case of Tschebotareff''s density theorem. [Copyright &y& Elsevier]

The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this paper is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. [Copyright &y& Elsevier]

Abstract: In 1988 Manickam and Singhi conjectured that for every positive integer and every , every set of real numbers whose sum is non-negative contains at least subsets of size whose sums are non-negative. In this paper we make use of Hall’s matching theorem in order to study some numbers related to this conjecture. [Copyright &y& Elsevier]

This paper addresses several questions related to the Hodge conjecture. First of all we consider the question, asked by Maillot and Soulé, whether the Hodge conjecture can be reduced to the case of varieties defined over number fields. We show that this is the case for the Hodge classes whose corresponding Hodge locus is defined over a number field. We also give simple criteria for this last condition to be satisfied. Finally we discuss the relation between this condition and the notion of absolute Hodge class. [ABSTRACT FROM AUTHOR]

Abstract: The main result of this paper is the following: Any minimal counterexample to Hadwiger''s Conjecture for the k-chromatic case is -connected. This improves the previous known bound due to Mader [W. Mader, Über trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243–252], which says that any minimal counterexample to Hadwiger''s Conjecture for the k-chromatic case is 7-connected for . This is the first result on the vertex connectivity of minimal counterexamples to Hadwiger''s Conjecture for general k. Consider the following problem: There exists a constant c such that any ck-chromatic graph has a -minor. This problem is still open, but together with the recent result in [T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces large complete bipartite graph minors, preprint], our main result implies that there are only finitely many minimal counterexamples to the above problem when . This would be the first step to attach the above problem. We also prove that the vertex connectivity of minimum counterexamples to Hadwiger''s Conjecture is at least -connected. This is also the first result on the vertex connectivity of minimum counterexamples to Hadwiger''s Conjecture for general k. [Copyright &y& Elsevier]

Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset , if , then must contain an antichain of cardinality . The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin’s Maximum and holds for all singular above the first strongly compact cardinal. [Copyright &y& Elsevier]

Tree is one of the most studied and practically useful classes of graphs and is the attention of a great number of studies. There is absence of generalized results for tree as a class and even for one kind of labeling as whole. Only specialized results exist limited to specific types of trees only. A number of conjectures stand being unsolved. Graham and Sloane (1980) conjectured trees to be Harmonious and Ringel‐Kotzig conjectured trees to be Graceful about three decades ago. Kotzig and Rosa (1970) ask the question whether all trees are Magic or not. No generalized result for Antimagic labeling is given for trees so far. This paper presents the methodologies to obtain the major labeling schemes for trees viz., Harmonious, Sequential, Felicitous, Graceful, Antimagic and found the trees to be not Magic except T(2,1), thus solving the said conjectures. These findings could also be useful for those working in fields where graphs serve as models. [ABSTRACT FROM AUTHOR]

We use the class of representation-finite algebras to investigate the finitistic dimension conjecture. In this way we obtain a large class of algebras for which the finitistic dimension conjecture holds. The main results in this paper are: (1) Let A be an artin algebra and let Ij,1⩽j⩽n be a family of ideals in A with I1I2⋯In=0, such that proj.dim(AIj)<∞ and proj.dim(Ij)A=0 for all j⩾3. If A/I1 and A/I2 are representation-finite and if A/Ij has finite finitistic dimension for j⩾3, then the finitistic dimension of A is finite. In particular, the finitistic dimension conjecture is true for algebras obtained from representation-finite algebras by forming dual extensions, trivially twisted extensions, Hochschild extensions, matrix algebras and tensor products with algebras of radical-square-zero. (2) Let A,B and C be three artin algebras with the same identity such that (i) C⊆B⊆A, and (ii) the Jacobson radical of C is a left ideal of B and the Jacobson radical of B is a left ideal of A. If A is representation-finite, then C has finite finitistic dimension. We also provide a way to construct algebras satisfying all conditions in (2), and this leads to a new reformulation of the finitistic dimension conjecture. [Copyright &y& Elsevier]

Let ai,j(n) denote the number of walks in n steps from (0,0) to (i,j), with steps (±1,0) and (0,±1), never touching a point (−k,0) with k⩾0 after the starting point. Bousquet-Mélou and Schaeffer conjectured a closed form for the number a−i,i(2n) when i⩾1. In this paper, we prove their conjecture, and give a formula for a−i,i(2n) for i⩽−1. [Copyright &y& Elsevier]

A tree T is k-centred graceful if it has a graceful labelling f such that f assigns the label k to the centre vertex (or one of the centres if the tree has odd diameter); similarly, a graph G is k-ubiquitously graceful if for every vertex v∈V(G) there is a graceful labelling f of G such that f(v)=k. In this paper we isolate a small and easily characterized subset of trees that are not 0-centred graceful, and a larger but still very manageable set of non-0-ubiquitously graceful trees; these we denote by D and D′, respectively. It is shown that all trees of diameter &les;4 that are not in D are 0-centred graceful, and all that are not in D′ are 0-ubiquitously graceful. Upon consideration of some very intriguing empirical data we conjecture that these results in fact extend to all trees. [Copyright &y& Elsevier]

Given two graphs G and H, an H-matching of G (or a tiling of G with H) is a subgraph of G consisting of vertex-disjoint copies of H. For an r-chromatic graph H on h vertices, we write u=u(H) for the smallest possible color-class size in any r-coloring of H. The critical chromatic number of H is the number χcr(H)=(r−1)h/(h−u). A conjecture of Komlo´s states that for every graph H, there is a constant K such that if G is any n-vertex graph of minimum degree at least (1−(1/χcr(H)))n, then G contains an H-matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all sufficiently large values of n when H is a 3-chromatic graph. [Copyright &y& Elsevier]

The 3x+1 map T is defined on the 2-adic integers by T(x)=x/2 for even x and T(x)=(3x+1)/2 for odd x and the 3x+1 conjecture states that the T-orbit of any positive integer contains 1. We define and study properties of the unique nontrivial autoconjugacy Ω of T. This autoconjugacy sends x to the unique 2-adic integer whose parity vector is the one's complement of the parity vector of x. We prove that if Ω maps rational numbers to rational numbers then there are no divergent T-orbits of positive integers. The map Ω is then used to restate the 3x+1 conjecture in a parity neutral form. We derive a necessary and sufficient condition for a cycle to be self-conjugate and show that self-conjugate cycles contain only positive elements. It is then shown that the only self-conjugate cycle of integers is {1,2}. Finally, we prove that for any rational 2-adic integer x, lim¯κn(x)/n+limκn(Ω(x))/n=1 where κn(x) is the number of ones in the first n digits of the parity vector of x, and we use this along with generalizations of known restrictions on lim¯ κn(x)/n to prove most of the results in the paper. [Copyright &y& Elsevier]

Gut and Spa˘taru (J. Math. Anal. Appl. 248 (2000) 233–246) proved a precise asymptotic theorem for random variables in the normal domain of attraction of some stable distribution with exponent α, 1<α&les;2. They also conjectured that the theorem should hold for the general case of the domain of attraction of a stable distribution. In this paper, we shall provide an example to deny their conjecture and study some further properties. [Copyright &y& Elsevier]

Let p be a prime, u be a linear recurring sequence of integers of order d and let S=3d2+9d/2+1. The main result of the present paper: if u is uniformly distributed mod pS, then it is uniformly distributed mod ps for all s&ges;1. This solves a longstanding folklore conjecture. [Copyright &y& Elsevier]

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on C n is known to be a K(p,1) space for the corresponding Artin group $\Cal A$. A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type ? n and its associated Artin group, the affine braid group $\tilde{\Cal A}$. Using the fact that $\tilde{\Cal A}$ can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, $n$-dimensional K(p,1)-space for $\tilde{\Cal A}$, and use it to prove the K(p,1) conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure. [ABSTRACT FROM AUTHOR]

The total domination number γt of a graph G=(V,E) is the minimum cardinality of a dominating set D of G such that every vertex of V has at least one neighbor in D. The least domination number γL of G is the minimum cardinality of a dominating set X of G whose domination number is the minimum. In this paper, we prove the following conjecture due to Odile Favaron:Conjecture. For any tree T, we have γL(T)/γt(T)&les;3/2. [Copyright &y& Elsevier]

This paper presents some known results on the approximate and null controllability of the Navier­Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions. [ABSTRACT FROM AUTHOR]

Let A2(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c∈(0,1), such that whenever &z.sfnc;f(z)&z.sfnc;⩽&z.sfnc;g(z)&z.sfnc; (f,g∈A2(D)) in the annulus c<&z.sfnc;z&z.sfnc;<1, then &z.dfnc;f&z.dfnc;⩽&z.dfnc;g&z.dfnc;. In 1999 Hayman proved Korenblum''s conjecture. But the sharp value of c (we use γ to denote this sharp value) is still unknown. In this paper we give an upper bound on γ, that is, γ<0.67795, which improves an earlier result of the author. [Copyright &y& Elsevier]

In the paper, we solve one conjecture on an inequality involving digamma function, an open problem, and a conjecture on monotonicity of functions involving generalized digamma function. We also prove a new inequality for digamma function. [ABSTRACT FROM AUTHOR]

The article presents information on a nearly 4-century-old problem called the Kepler conjecture, which is solved by Thomas Hales. The editor of "Annals of Mathematics," one of the most prestigious journals in mathematics, invited him to submit his proof to Annals. Over a period of 4 years, a team of 12 referees wrestled with the lengthy paper and eventually raised a white flag. They informed the editor that they were only "99 percent" certain that it was correct. In particular, they could not vouch for the validity of the lengthy computer calculations that were essential to Hales's proof. INSET: Have a Coq and a Smile.

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. In this paper we prove that for each n&ges;6, there exists a 4-coloured tournament Tn of order n satisfying the two following conditions: (1) Tn does not contain C3 (the directed cycle of length 3, whose arcs are coloured with three distinct colours), and (2) Tn does not contain any vertex v such that for every other vertex x of Tn, there is a monochromatic directed path from x to v. This answers a question proposed by Shen Minggang in 1988. [Copyright &y& Elsevier]