A generalization of Yager’s f-generated implications

Since Yager introduced f-generated implications in 2004, this class of fuzzy implications has been extensively investigated. In this paper, we generalize f-generated implications and get a new class of fuzzy implications called (f, g)-implications, which is different from the usual known classes of fuzzy implications. We discuss the basic algebraic properties of (f, g)-implications and study some classical logic tautologies (i.e., law of importation, contrapositive symmetry and distributivity over t-norms or t-conorms) for (f, g)-implications. Characterization of solutions to the corresponding fuzzy functional equations is obtained.     

Double reduction of a nonlinear (2+1) wave equation via conservation laws

Conservation laws of a nonlinear (2+1) wave equation u_tt = (f(u)u_x)_x +  (g(u)u_y)_y involving arbitrary functions of the dependent variable are obtained, by writing the equation in the partial Euler-Lagrange form. Noether-type operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary functions. If either of f(u) or g(u) is an arbitrary nonconstant function, we show that there are an infinite number of conservation laws. If both f(u) and g(u) are arbitrary nonconstant functions, it is shown that there exist infinite number of conservation laws when f′(u) and g′(u) are linearly dependent, otherwise there are eight conservation laws. Finally, we apply the generalized double reduction theorem to a nonlinear (2+1) wave equation when f′(u) and g′(u) are linearly independent.     

Lack of relative monotonicity among various measures of trihedral angles

We consider the various measures on trihedral angles that have appeared in the literature and we show that no two of these measures are monotone with respect to each other. In other words, for any measures f, g, there exist trihedral angles ??, ??, ??, ?? such that f(??) >? f(??), g(??) <? g(??), f(??) >? f(??), g(??) >? g(??). This is done through an elementary and systematic method based on multivariable calculus.     

The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=R^m_￥(g)=R(g)-2D_glog f-|?_glog f|_g²{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N ⁿ , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS¹{\mathbb{R}\times\mathbb{S}^1}.     

Spinor equivalence of quadratic forms

Let f be an integral quadratic form in three or more variables and g any form in the genus of f. There exist an effectively determinable prime p and a form g′, belonging to the proper spinor genus of g, such that g′ is a p-neighbor of f in the graph of f. Using this, an alternative decision procedure for the spinor equivalence of quadratic forms is given.     

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E₈ lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ?x?) can be written as f=gn for g∈R, n?2. Let Pn:={gn|g∈R} and let . We show among other things that (i) for f∈R, f∈Pn⇔f (mod μn)∈Pn, and (ii) if f∈Pn, there is a unique g∈Pn with coefficients mod μn/n such that f≡gn (mod μn). In particular, if f≡1 (mod μn) then f∈Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E₈ in R⁸) is in Pn if n is of the form i²j³k⁵ (i?3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length m² is in Pr² (and similarly that the theta series of the Barnes-Wall lattice BWm² is in Pm²). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈Pn (n?2) with coefficients restricted to the set {1,2,…,n}.     

An invariance of geometric mean with respect to Lagrangian means

The invariance of the geometric mean G with respect to the Lagrangian mean-type mapping (Lf,Lg), i.e. the equation G○(Lf,Lg)=G, is considered. We show that the functions f and g must be of high class regularity. This fact allows to reduce the problem to a differential equation and determine the second derivatives of the generators f and g.     

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ(fg) of the product of any two elements f and g in A coincides with the spectrum σ(TfTg). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ(TfTg)=σ(fg) holds for every f,g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ(TfTg)⊂σ(fg) holds for every f,g.     

Hankel matrices and the infinite companion

A formula of Barnett type relating the Bezoutian B(f,g) to the Hankel matrix H(g/f) is extended to rectangular Bezoutians. The proof is based on an interesting relation between the family of all Hankel matrices corresponding to the Markov parameters of g/f and the infinite companion matrix corresponding to f.     

Density condensation of Boolean formulas

The following problem is considered: given a Boolean formula f, generate another formula g such that: (i) If f is unsatisfiable then g is also unsatisfiable. (ii) If f is satisfiable then g is also satisfiable and furthermore g is “easier” than f. For the measure of this easiness, we use the density of a formula f which is defined as (the number of satisfying assignments)/2ⁿ, where n is the number of Boolean variables of f. In this paper, we mainly consider the case that the input formula f is given as a 3-CNF formula and the output formula g may be any formula using Boolean AND, OR and negation. Two different approaches to this problem are presented: one is to obtain g by reducing the number of variables and the other by increasing the number of variables, both of which are based on existing SAT algorithms. Our performance evaluation shows that, a little surprisingly, better SAT algorithms do not always give us better density-condensation algorithms.     

Picard-Lefschetz monodromy of products

Let f and g be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of fg to that of f and g separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu.     

A generalization of the cosine-sine functional equation on groups

The functional equation f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y) is solved where f, g, h are complex functions defined on a group.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

Riemann integrability and Lebesgue measurability of the composite function

If f is continuous on the interval [a,b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [α,β] and g([α,β])⊂[a,b], then f○g is Riemann integrable (resp. measurable) on [α,β]. A well-known fact, on the other hand, states that f○g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a c²-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f∈V?{0} and g∈W?{0}, f○g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g∈W?{0} there exists a c-dimensional space V of measurable functions such that f○g is not measurable for all f∈V?{0}.     

Commuting Toeplitz operators on the Segal-Bargmann space

Consider two Toeplitz operators Tg, Tf on the Segal-Bargmann space over the complex plane. Let us assume that g is a radial function and both operators commute. Under certain growth condition at infinity of f and g we show that f must be radial, as well. We give a counterexample of this fact in case of bounded Toeplitz operators but a fast growing radial symbol g. In this case the vanishing commutator [Tg,Tf]=0 does not imply the radial dependence of f. Finally, we consider Toeplitz operators on the Segal-Bargmann space over Cn and n>1, where the commuting property of Toeplitz operators can be realized more easily.     

On the representation of functions as a sum of derivatives

We show that there is an integrable function g of two variables which cannot be represented as a sum g = f₀ + ∂f₁ + ∂₂f₂, where f₀,f₁,f₂ are functions with integrable gradient.     

What Is the Complexity of Stieltjes Integration?

We study the complexity of approximating the Stieltjes integral ∫¹₀ f(x) dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show that r(n)≍n^{−min{r, s+1}} and that comp(ε)≍ε^{−1/min{r, s+1}}. We also present an algorithm that computes an ε-approximation at nearly minimal cost.     

On zeros and fixed points of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+1)−f(z). A number of results are proved concerning the existences of zeros and fixed points of g(z) or g(z)/f(z) which expand results of Bergweiler and Langley [W. Bergweiler, J.K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007) 133-147].     

Rigidity and stability of the Leibniz and the chain rule

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T ₁, T ₂,A: C ^k (?) → C(?) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C ^k (?), namely, V (f · g) = (T ₁ f) · g + f · (T ₂ g) for k = 1, V (f · g) = (T ₁ f) · g + f · (T ₂ g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T ₁ f) ○ g · (T ₂ g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T ₁ f) · (Ag) + (Af) · (T ₂ g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T ₁ and T ₂ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.