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The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]

Let two nonperiodic binary morphisms ... be given. A word w is called a solution of g and h if g(w) = h(w). We say that a solution w is simple if whenever ... are prefixes of wu such that ... for some word z, then |u| = |u'| = k|w|, for some k ∈ N. In this paper we will study simple solutions and show that if a word w is a simple solution containing at least nine occurrences of the letter a and at least nine occurrences of the letter b, then either w = (ab)ia, or w = ajbk with gcd (j, k) = 1, up to the exchange of letters a and b. [ABSTRACT FROM AUTHOR]

In this paper, from the viewpoint of quandle construction we prove that for any link type L and any diagram of it, there exists another diagram of L such that these two diagrams are Ω1-dependent, Ω2-dependent and Ω3-dependent. [ABSTRACT FROM AUTHOR]

In this paper, we define and study a special type of trisections in a module category, namely the compact trisections which characterize quasi-directed components. We apply this notion to the study of laura algebras and we use it to define a class of algebras with predictable Auslander-Reiten components. [ABSTRACT FROM AUTHOR]

The classification of nilpotent Lie algebras of maximal rank was solved by Santharoubane. In the present paper, we prove that the classification of non-degenerate solvable Lie algebras of maximal rank can be obtained from the work of Santharoubane. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. [ABSTRACT FROM AUTHOR]

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism $\sigma_E^*$. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), $ht(\phi_E|_{\mathcal A_E})$, hl(E) and hb(E) all coincide, where $\mathcal A_E$ is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, $h_l(E)\leq ht(\phi_E|_{\mathcal A_E})$ has been known recently, and here we show that $ht(\phi_E|_{\mathcal A_E})\leq h_b (E^t)$, where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras $\mathcal A_E(v)$ of $\mathcal A_E$ are examined to prove that $h_l(E)\leq ht(\phi_E|_{\mathcal A_E(v)})$ for each vertex v. [ABSTRACT FROM AUTHOR]

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d. [ABSTRACT FROM AUTHOR]

Reverse nearest neighbor queries are defined as follows: Given an input point set P, and a query point q, find all the points p in P whose nearest point in P ∪ {q} \ {p} is q. We give a data structure to answer reverse nearest neighbor queries in fixed-dimensional Euclidean space. Our data structure uses O(n) space, its preprocessing time is O(n log n), and its query time is O(log n). [ABSTRACT FROM AUTHOR]

We revisit the harmonic oscillator algebra in an indefinite metric by reinterpreting consistently the time-like components in a way compatible with a positive-definite metric. We show that, despite its unusual features, the relativistic oscillator algebra can be derived starting from the Euclidean q-oscillator algebra. The consistency of this isomorphism is examined at different levels including the possible implication on the dynamics of the two formalisms by means of their respective Hamiltonians. [ABSTRACT FROM AUTHOR]

Let L be a degree-2L-function associated to a Maass cusp form. We explore an algorithm that evaluates t values of L on the critical line in time O(t1+𝜖). We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution. [ABSTRACT FROM AUTHOR]

Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s2(n) = s2(n2). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case p(n) = n2, looking at the subsets of n where sq(n) = sq(n2) = k for fixed k. [ABSTRACT FROM AUTHOR]

We solve the equation f(x, y)3 + g(x, y)3 = x3 + y3 for homogeneous f, g ∈ ℂ(x, y), completing an investigation begun by Viète in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring isomorphic to ℤ[e2πi/3]. [ABSTRACT FROM AUTHOR]

We establish the rate of decay to 0 and we study the oscillation properties of solutions to the scalar second order ODE: u″ + c|u′|αu′ + |u|βu = 0, where c, α, β are positive constants. Various extensions (forced equation, system) are considered. [ABSTRACT FROM AUTHOR]

A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is An - 1. The proof involves a diagrammatic version of the Brauer algebra of type Dn of which the generalized Temperley–Lieb algebra of type Dn is a subalgebra. [ABSTRACT FROM AUTHOR]

Choice functions play an important role in political science. We present structure results of choice functions following an algebraic approach. It is logical that rational political actors are not able to perceive every alternative in a set of alternatives X. They may also choose to exclude some alternatives from consideration. Let S be a subset of X that some set of political actors are not able to perceive. Then the complement X\S is the set comprising those alternatives considered in the decision process. If C is a choice function on X, then C(X\S) are the alternatives chosen and X\C(X\S) are those that are not. We are thus led to the concept of an upper choice function. We define upper choice functions in terms of choice functions and we give relationships between them. [ABSTRACT FROM AUTHOR]

We prove that the Weak Reflection Principle does not imply that every stationary set reflects to an internally approachable set. We show that several variants of internal approachability, namely, internally unbounded, internally stationary, and internally club, are not provably equivalent. [ABSTRACT FROM AUTHOR]

Richard Thompson's group F is nonamenable if and only if there exists a ruinous set. It is known that no three element set is ruinous. Here we explicitly construct a family of thin sets for any three element set each of whose trees has three carets. We then show where this construction fails when each of the trees has four or more carets. [ABSTRACT FROM AUTHOR]

We investigate the function b(n) = ∑ 1, where the summing is over all i for which $(n,i) > 1, i | {\fontsize{9}{11}\selectfont\Big(\!\!\begin{array}{c}n-i-1\\[2pt]i-1\end{array}\!\!\Big)}$. [ABSTRACT FROM AUTHOR]

We construct a diagram ${\mathcal{D}_{\scriptscriptstyle{\bowtie}}}$, indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 24, such that for any variety V of algebras, if ${\mathcal{D}_{\scriptscriptstyle{\bowtie}}}$ has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M3. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive Using earlier work of K. A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity. In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition Con Φ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem. [ABSTRACT FROM AUTHOR]

We consider nonlinear generalizations of fractal interpolating functions applied to functions of one and two variables. The use of such interpolating functions in representing images by an iterated function system is illustrated, particularly as applied to the problem of resizing images, or portions thereof. [ABSTRACT FROM AUTHOR]

Recent developments of Baxter algebras have led to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first and the second kinds, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras. [ABSTRACT FROM AUTHOR]