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We characterize the completely determined Borel subsets of HYP as exactly the Δ 1 (L ω 1 c k ) subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. [ABSTRACT FROM AUTHOR]

Let (S , τ) be a nontrivial compact metric space with metric τ and g : S → S be a continuous self-map, ℬ (S) be the sigma-algebra of Borel subsets of S , and ν be a Borel probability measure on (S , ℬ (S)) with ν (V) ∈ (0 , + ∞) for any open subset V ≠ ∅ of S. This paper proves the following results : (1) If the pair (g , ν) has the property that for any β > 0 , there is r (β) > 0 such that ν p ∈ S : 1 m ∑ i = 0 m − 1 V (g i (p)) − ∫ V d ν > β ≤ e − m r (β) , for any open subset V ≠ ∅ of S and all n ≥ 1 sufficiently large (where V is the characteristic function of the set V), then the following hold : (a) The map g is topologically ergodic. (b) The upper density μ ¯ (N g (p , V)) of N g (p , V) is positive for any open subset V ≠ ∅ of S , where N g (p , V) = { m ∈ { 0 , 1 , ... } : g m (p) ∈ V }. (c) There is a g -invariant Borel probability measure ν having full support (i.e. supp (ν) = S). (d) Sensitivity of the map g implies positive lower density sensitivity, hence ergodical sensitivity. (2) If μ ̲ (N g (A , B)) = 1 for any two nonempty open subsets A , B , then there exists λ > 0 satisfying μ ̲ (N g (C , λ)) = 1 for any nonempty open subset C ⊂ S , where N g (C , λ) = { l ∈ { 0 , 1 , ... } : there exist a , b ∈ C with τ (g l (a) , g l (b)) > λ }. [ABSTRACT FROM AUTHOR]

In [S. M. Hegde, Set colorings of graphs, European J. Combin.30 (2009) 986–995.] Hegde introduced the notion of set colorings of a graph G as an assignment of distinct subsets of a finite set X of n colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the subsets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G are the set of all nonempty subsets of X , then the coloring is said to be a strong set-coloring and G is said to be strongly set-colorable. In this paper, we report some new necessary conditions and propose a conjuncture for the sufficient condition for a graph to admit strong set-coloring. We also identify and characterize some new classes of graphs admitting strong set-coloring. In addition to these, we also propose strategies to construct infinite families graphs admitting strong set-coloring. [ABSTRACT FROM AUTHOR]

A subset T of vertices in a hypergraph H is a transversal if T has a nonempty intersection with every edge of H. The transversal number of H is the minimum size of a transversal in H. A subset S of vertices in a graph G with no isolated vertex, is a total dominating set if every vertex of G is adjacent to a vertex of S. The minimum cardinality of a total dominating set in G is the total domination number of G. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph. [ABSTRACT FROM AUTHOR]

A subset of ℝd is called “sticky” if it cannot be isotoped off of itself by a small ambient isotopy. Sticky wild Cantor sets are constructed in ℝd for each d≥4. [ABSTRACT FROM AUTHOR]

Let be a formation, be a partition of the set of all primes into mutually disjoint subsets and be the formation of all -groups from . We found under what conditions the intersection of normalizers of all -maximal subgroups for all coincides with the -hypercenter of a finite group. As corollaries we obtained well-known results of R. Baer about the hypercenter of a finite group. [ABSTRACT FROM AUTHOR]

We introduce a method to reduce to the real case the calculus of the box-counting dimension of subsets of the unit cube , . The procedure is based on the existence of special types of -dense curves (a generalization of the space-filling curves) in called -uniform curves. [ABSTRACT FROM AUTHOR]

Let and . A ring is called a -ring if for every -subsets of there exist and such that . We give noncommutative examples of such rings, and we discuss finiteness and commutativity. [ABSTRACT FROM AUTHOR]

We prove commutativity of rings with 1 which satisfy certain th power identities on proper subsets of . [ABSTRACT FROM AUTHOR]

In this paper, we study open questions on Ćirić type theorems proposed by Petruşel.1 First, we prove two theorems similar to results of L. B. Ćirić2 and I. A. Rus3 which are positive answers to the question presented in Remark 2.3 of Petruşel's paper.1 Then we construct a counterexample to illustrate obtained results and also to disprove results similar to Theorems 4.1 and 4.2 for a multi-fractal map generated by an iterated multi-valued system of multi-valued maps of Ćirić type, which is a negative answer to the question presented on page 243 of Petruşel's paper.1 Our approach is to apply the argument for a metric to a distance between two bounded subsets of a metric space. This approach differs from the selection approach known in the literature. [ABSTRACT FROM AUTHOR]

This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set can be affinely embedded into a symmetric Cantor set , then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062-1079]. [ABSTRACT FROM AUTHOR]

Bouncing robots are mobile agents with limited sensing capabilities adjusting their movements upon collisions either with other robots or obstacles in the environment. They behave like elastic particles sliding on a cycle or a segment. When two of them collide, they instantaneously update their velocities according to the laws of classical mechanics for elastic collisions. They have no control on their movements which are determined only by their masses, velocities, and upcoming sequence of collisions. We suppose that a robot arriving for the second time to its initial position dies instantaneously. We study the survivability of collections of swarms of bouncing robots. More exactly, we are looking for subsets of swarms such that after some initial bounces which may result in some robots dying, the surviving subset of the swarm continues its bouncing movement, with no robot reaching its initial position. For the case of robots of equal masses and speeds we prove that all robots bouncing in the segment must always die while there are configurations of robots on the cycle with surviving subsets. We show the smallest such configuration containing four robots with two survivors. We show that any collection of less than four robots must always die. On the other hand, we show that robots always die where (and ) is the number of robots starting their movements in clockwise (respectively, counterclockwise) direction in swarm . When robots bouncing on a cycle or a segment have arbitrary masses we show that at least one robot must always die. Further, we show that in either environment it is possible to construct swarms with survivors. We prove, however, that the survivors in the segment must remain static indefinitely while in the case of the cycle it is possible to have surviving collections with strictly positive kinetic energy. [ABSTRACT FROM AUTHOR]

For a graph , a Roman dominating function (RDF) has the property that every vertex with has a neighbor with . The weight of a RDF is the sum , and the minimum weight of a RDF on is the Roman domination number of . The Roman bondage number of is the minimum cardinality of all sets for which . A graph is in the class if the Roman domination number remains unchanged when a vertex is deleted. In this paper, we obtain tight upper bounds for and provided a graph is in . We present necessary and sufficient conditions for a tree to be in the class . We give a constructive characterization of -trees using labelings. [ABSTRACT FROM AUTHOR]

Explicit solutions of the cubic Fermat equation are constructed in ring class fields , with conductor prime to , of any imaginary quadratic field whose discriminant satisfies (mod ), in terms of the Dedekind -function. As and vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension of the -adic field . This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner's conjecture: the cubic Fermat equation has a nontrivial solution in if (mod ) and the class number is not divisible by . If , congruence conditions for the trace of specific elements of are exhibited which imply the existence of a point of infinite order in . [ABSTRACT FROM AUTHOR]

We propose a simple model of landfill and study a minimal time control problem where the re-circulation leachate is the manipulated variable. We propose a scheme to construct the optimal strategy by dividing the state space into three subsets , and the complementary. On and , the optimal control is constant until reaching target, while it can exhibit a singular arc outside these two subsets. Moreover, the singular arc could have a barrier. In this case, we prove the existence of a switching curve that passes through a point of prior saturation under the assumption that the set intersects the singular arc. Numerical computations allow then to determine the switching curve and depict the optimal synthesis. [ABSTRACT FROM AUTHOR]

A two-valued function defined on the vertices of a graph , is a majority dominating function if the sum of its function values over at least half the closed neighborhoods is at least one. That is, for at least half the vertices , , where consists of and every vertex adjacent to . The majority domination number of a graph , denoted , is the minimum value of over all majority dominating functions of . The majority reinforcement number of , denoted by , is defined to be the minimum cardinality of a set of edges such that . In this paper, we initiate the study of majority reinforcement number and determine the exact values of for paths and cycles. [ABSTRACT FROM AUTHOR]

We show that all knots up to six crossings can be represented by polynomial knots of degree at most , among which except for and all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any -crossing knot in degree ? In this paper we try to partially answer this question. For an integer , we define a set to be the set of all polynomial knots given by such that and . This set can be identified with a subset of and thus it is equipped with the natural topology which comes from the usual topology . In this paper we determine a lower bound on the number of path components of for . We define a path equivalence for polynomial knots in the space and show that it is stronger than the topological equivalence. [ABSTRACT FROM AUTHOR]

Let G be a group. A subset X of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y. If |X| ≥ |Y| for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, the bound for the cardinality of a maximal non-commuting set in a finite p-group G is determined, where G is a non-abelian p-group given by a central extension as 1 → ℤpm → G → ℤp× ⋯ × ℤp → 1 and its derived subgroup has order p. [ABSTRACT FROM AUTHOR]

In the first half, we present a generalization of the graph theory prime-number theorem due to Audrey Terras et al. In the last half, we study relations between several densities of a subset consisting of primes in a graph, and by using the first result, we compute a distribution of a subset. This is an answer for the "research problem" in the page 197 of Terras' textbook "Zeta Functions of Graphs." [ABSTRACT FROM AUTHOR]

In previous work, two axiomatic characterizations were given for the median function on median graphs: one involving the three simple and natural axioms anonymity, betweenness and consistency; the other involving faithfulness, consistency and -Condorcet. To date, the independence of these axioms has not been a serious point of study. The aim of this paper is to provide the missing answers. The independent subsets of these five axioms are determined precisely and examples provided in each case on arbitrary median graphs. There are three cases that stand out. Here nontrivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout. [ABSTRACT FROM AUTHOR]

Performance improvements to the known opacity correction mechanisms for over-sampled volume ray casting (VRC), especially using two forms of commodity hardware, are explored. Data-parallel strategies that enable exploitation of parallelism using either: (1) a programmable graphics processing unit (GPU) or (2) cluster computation are a prime focus. The GPU-based approach is finely granular. The cluster-based approaches here utilize less finely granular processing that allows acceleration through multi-processing and multi-threading. These approaches also include features, such as early ray termination, empty-space skipping, term rearrangement, and term reduction, that have not been previously explored in depth for opacity-corrected VRC. A new strategy enabling more accurate opacity correction is also presented. The performance of the improvements on real volume data are also explored. The improvements allow opacity correction to be performed in a way that efficiently exploits either GPU- or cluster-based capabilities. [ABSTRACT FROM AUTHOR]