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Property for an arbitrary complex, Fano manifold is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of . Conjecture is a conjecture that property holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture holds for two more classes and an example in a third class of Pasquier’s list. Perron–Frobenius theory reduces our proofs to be graph-theoretic in nature.
We study a modified notion of Ollivier’s coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than 1. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive exact Ricci curvature formulas for strongly regular graphs of girths 4 and 5 using elementary means. An example is provided that shows there is no exact formula for the Ricci curvature for strongly regular graphs of girth 3 that is purely in terms of graph parameters.
An equidistant polytope is a special equidistant set in the space all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in the interior of the convex hull of the other one, we can speak about inner and outer focal points of the equidistant polytope. It is of type , where is the number of the outer focal points and is the number of the inner focal points. The equidistancy is the generalization of convexity because a convex polytope can be given as an equidistant polytope of type , where . We present some general results about the basic properties of the equidistant polytopes: convex components, graph representations, connectedness, correspondence to the Voronoi decomposition of the space etc. In particular, we are interested in equidistant polytopes of dimension (equidistant polygons). Equidistant polygons of type will be characterized in terms of a constructive (ruler-and-compass) process to recognize them. In general they are pentagons with exactly two concave angles such that the vertices at which the concave angles appear are joined by an inner diagonal related to the adjacent sides of the polygon in a special way via the three reflections theorem for concurrent lines. The last section is devoted to some special arrangements of the focal points to get the concave quadrangles as equidistant polygons of type .
The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values for any polynomial which is not equivalent to a Dickson polynomial.
First, we consider the problem of hedging in complete binomial models. Using the discrete-time Föllmer–Schweizer decomposition, we demonstrate the equivalence of the backward induction and sequential regression approaches. Second, in incomplete trinomial models, we examine the extension of the sequential regression approach for approximation of contingent claims. Then, on a finite probability space, we investigate stability of the discrete-time Föllmer–Schweizer decomposition with respect to perturbations of the stock price dynamics and, finally, perform its asymptotic analysis under simultaneous perturbations of the drift and volatility of the underlying discounted stock price process, where we prove stability and obtain explicit formulas for the leading-order correction terms.
We explore free knot diagrams, which are projections of knots into the plane which don’t record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability. Every free knot diagram is proven to produce trefoil knots, and certain simple families of free knots are completely worked out. We make some conjectures (supported by computer-generated data) about bounds on the probability of a knot arising from a fixed free diagram being the unknot, trefoil, or figure-eight knot.
The conjugation diameter of a group is the largest diameter of its Cayley graphs with respect to conjugation-invariant generating sets. It is a strong form of the extensively studied concept of the diameter of . We compute the conjugation diameter of the symmetric groups.
A second-order discrete boundary value problem with mixed periodic boundary conditions is studied. Sufficient conditions on the multiplicity of solutions in a weak sense are obtained by using the critical point theory. An example is given to demonstrate the applications of our results as well.
We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of and have 11 and 12 vertices, respectively. In general, we show that (resp. ) vertices suffice to obtain a flag triangulation of the connected sum of copies of (resp. ). In dimension 3, we describe an algorithm based on the Lutz–Nevo theorem which provides supporting computational evidence for the following generalization of the Charney–Davis conjecture: for any flag 3-manifold, , where is the number of -dimensional faces and is the first Betti number over a field . The conjecture is tight in the sense that for any value of , there exists a flag 3-manifold for which the equality holds.
We introduce three new fractional Gompertz difference equations using the Riemann–Liouville discrete fractional calculus. These three models are based a nonfractional Gompertz difference equation, and they differ depending on whether a fractional operator replaces the difference operator, the integral operator defining the logarithm, or both simultaneously. An explicit solution to one of them is achieved with restricted parameters and recurrence relation solutions are derived for all three. Finally, we fit these models to data to compare them with a previously published discrete fractional Gompertz model and the continuous model.
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