A bijection between Littlewood-Richardson tableaux and rigged configurations

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product .     

A connection between Fano configurations and Minkowski planes of odd order

It is shown in this paper that a pair of points contained in a Fano configuration in a projective plane of odd order cannot induce a Minkowski plane. From this result we derive that no pair of points in the Hughes plane of order 9 can induce a Minkowski plane.     

On the several differences between primes

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,4(5 × 10¹⁰) = 118905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computedB ₄(5×10¹⁰)=1.197054473029 andB ₄=1.197054±7×10^?6. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,6(5 × 10¹⁰) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computedB ₆(5×10¹⁰)=0.93087506039231 andB ₆=1.135835±1.2×10^?6.     

An algorithm for calculating logarithms through displacement

A procedure is proposed for the factorial expansion of special real numbers, each of which in turn allows a similar expansion. Problems of executing the program for calculating a natural logarithm are discussed. Approaches to generalizing the method are considered.     

ESG score prediction through random forest algorithm

Computational Management Science - Environment-related risks affect assets in various sectors of the global economy, as well as social and governance aspects, giving birth to what is known as ESG...     

On configurations of subspaces of a Hilbert space with fixed angles between them

We investigate the set of irreducible configurations of subspaces of a Hilbert space for which the angle between every two subspaces is fixed. This is the problem of *-representations of certain algebras generated by idempotents and depending on parameters (on the set of angles). We separate the class of problems of finite and tame representation type. For these problems, we indicate conditions on angles under which the configurations of subspaces exist and describe all irreducible representations.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 606–615, May, 2004.     

A metric space to study differences between polynucleotides

We introduce a new metric in the space of fuzzy polynucleotides.     

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

Implementation of fractional-order electromagnetic potential through a genetic algorithm

Several phenomena present in electrical systems motivated the development of comprehensive models based on the theory of fractional calculus (FC). Bearing these ideas in mind, in this work are applied the FC concepts to define, and to evaluate, the electrical potential of fractional order, based in a genetic algorithm optimization scheme. The feasibility and the convergence of the proposed method are evaluated.     

Some differences between difference equations and differential equations

In this expository paper we discuss some instances in which analogues which might be expected between the behavior of solutions of differential equations and difference equations fail to hold, focussing particularly on questions related to boundedness and oscillation.     

Convergence analysis of a proximal point algorithm for minimizing differences of functions

Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A significant progress to go beyond convexity was made by considering the class of functions representable as differences of convex functions. In this paper, we introduce a generalized proximal point algorithm to minimize the difference of a nonconvex function and a convex function. We also study convergence results of this algorithm under the main assumption that the objective function satisfies the Kurdyka–?ojasiewicz property.     

A robust descent type algorithm for geophysical inversion through adaptive regularization

A robust descent type algorithm through adaptive regularization is designed to solve a geophysical inverse problem. The scheme uses a regularized descent direction which is obtained through minimization of smoothed functional in every iterative step. The step length factor is chosen using Armijo's rule. Numerical experiment conducted to invert synthetic and field geophysical data demonstrates a high order of robustness in retrieving the model parameters.     

On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and η(G)=Sz(G)−W(G), where W(G) and Sz(G) are the Wiener and Szeged indices of G, respectively. A well-known result of Klav?ar, Rajapakse, and Gutman states that η(G)≥0, and by a result of Dobrynin and Gutman η(G)=0 if and only if each block of G is complete. In this paper, a path-edge matrix for the graph G is presented by which it is possible to classify the graphs in which η(G)=2. It is also proved that there is no graph G with the property that η(G)=1 or η(G)=3. Finally, it is proved that, for a given positive integer k,k≠1,3, there exists a graph G with η(G)=k.     

Fuzzy nonlinear programming for mixed-discrete design optimization through hybrid genetic algorithm

Many practical engineering optimization problems involve discrete or integer design variables, and often the design decisions are to be made in a fuzzy environment in which the statements might be vague or imprecise. A mixed-discrete fuzzy nonlinear programming approach that combines the fuzzy λ-formulation with a hybrid genetic algorithm is proposed in this paper. This method can find a globally compromise solution for a mixed-discrete fuzzy optimization problem, even when the objective function is nonconvex and nondifferentiable. In the construction of the objective membership function, an error from the early research work is corrected and the right conclusion has been made. The illustrative examples demonstrate that more reliable and satisfactory results can be obtained through the present method.     

An algorithm for the difference between set covers

A set cover for a set S is a collection C of special subsets whose union is S. Given covers A and B for two sets, the set-cover difference problem is to construct a new cover for the elements covered by A but not B. Applications include testing equivalence of set covers and maintaining a set cover dynamically. In this paper, we solve the set-cover difference problem by defining a difference operation A-B, which turns out to be a pseudocomplement on a distributive lattice. We give an algorithm for constructing this difference, and show how to implement the algorithm for two examples with applications in computer science: face covers on a hypercube, and rectangle covers on a grid. We derive an upper bound on the time complexity of the algorithm, and give upper and lower bounds on complexity for face covers and rectangle covers.     

Elementary differences between the degrees of unsolvability and degrees of compressibility

Given two infinite binary sequences A,B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A, modulo a constant. This relation, introduced in Nies (2005) [14] and denoted by A≤_LKB, is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes induced by ≤_LK are called LK degrees (or degrees of compressibility) and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K’ sets. A well-known result from Nies (2005) [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity.We show that with respect to ≤_LK, given any non-trivial sets X,Y there is a computably enumerable set A which is not K-trivial and it is below X,Y. This shows that the local structures of and Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the or structures of LK degrees. The proofs introduce a new technique of permitting below a set that is not K-trivial, which is likely to have wider applications.     

An explicit formula for differences between Laplacian-eigenvector components using coalesced graphs

We obtain an explicit formula for the absolute difference between two eigenvector components for a weighted graph’s Laplacian matrix, in terms of the Laplacian’s eigenvalues as well as the eigenvalues of matrices associated with certain coalesced graphs. We then briefly illustrate two uses of this formula, in analyzing graph modifications.