Path Theorem Research Articles

A Shuffle Theorem for Paths Under Any Line

AbstractWe generalize the shuffle theorem and its$(km,kn)$version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the$(km,kn)$Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whosexandyintercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of$\operatorname {\mathrm {GL}}_{l}$characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.

A Shuffle Theorem for Paths Under Any Line – Corrigendum

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On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index

In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented.

A Method for Automatic Inversion of Oblique Ionograms

In this study, a method is proposed to carry out automatic inversion of oblique ionograms to extract the parameters and electron density profile of the ionosphere. The proposed method adopts the quasi-parabolic segments (QPS) model to represent the ionosphere. Firstly, numerous candidate electron density profiles and corresponding vertical traces were, respectively, calculated and synthesized by adjusting the parameters of the QPS model. Then, the candidate vertical traces were transformed to oblique traces by the secant theorem and Martyn’s equivalent path theorem. On the other hand, image processing technology and characteristics of oblique echoes were adopted to automatically scale the key parameters (the maximum observable frequency and minimum group path, etc.) from oblique ionograms. The synthesized oblique traces, whose parameters were close to autoscaled parameters, were selected as the candidate traces to produce a correlation with measured oblique ionograms. Lastly, the proposed algorithm searched the best-fit synthesized oblique trace by comparing the synthesized traces with oblique ionograms. To test its feasibility, oblique ionograms were automatically scaled by the proposed method and these autoscaled parameters were compared with manual scaling results. The preliminary results show that the accuracy of autoscaled maximum observable frequency and minimum group path of the ordinary trace of the F2 layer is, respectively, about 91.98% and 86.41%, which might be accurate enough for space weather specifications. It inspires us to improve the proposed method in future studies.

The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $$S^{2n+1}/ \Gamma $$

Let $$M=S^{2n+1}/ \Gamma $$ , $$\Gamma $$ is a finite group which acts freely and isometrically on the $$(2n+1)$$ -sphere and therefore M is diffeomorphic to a compact space form. In this paper, we first investigate Katok’s famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form M, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form (M, F) when there exist only finitely many distinct contractible minimal closed geodesics on (M, F). As its applications, using this identity and the enhanced common index jump theorem for symplectic paths proved by Duan et al. (Calc Var PDEs 55(6):55–145, 2016), we show that there exist at least $$2n+2$$ distinct closed geodesics on every compact space form $$S^{2n+1}/ \Gamma $$ with a bumpy irreversible Finsler metric F under some natural curvature condition, which is the optimal lower bound due to Katok’s example.

A Chung–Feller theorem for lattice paths with respect to cyclically shifting boundaries

Irving and Rattan gave a formula for counting lattice paths dominated by a cyclically shifting piecewise linear boundary of varying slope. Their main result may be considered as a deep extension of well-known enumerative formulas concerning lattice paths from (0, 0) to (kn, n) lying under the line $$x=ky$$ (e.g., the Dyck paths when $$k=1$$ ). On the other hand, the classical Chung–Feller theorem tells us that the number of lattice paths from (0, 0) to (n, n) with exactly 2k steps above the line $$x=y$$ is independent of k and is therefore the Catalan number $$\frac{1}{n+1}{2n\atopwithdelims ()n}$$ . In this paper, we study the number of lattice path boundary pairs $$(P,\mathbf{a})$$ with k flaws, where P is a lattice path from (0, 0) to (n, m), $$\mathbf{a}$$ is a weak m-part composition of n, and a flaw is a horizontal step of P above the boundary $$\partial \mathbf{a}$$ . We prove bijectively, for a given $$\mathbf{a}$$ , that summing these numbers over all cyclic shifts of the boundary $$\partial \mathbf{a}$$ is equal to $${n+m\atopwithdelims ()m-1}$$ . That is, we generalize the Irving–Rattan formula to a Chung–Feller-type theorem. We also give a refinement of this result by taking the number of double ascents of lattice paths into account.

On extremal hypergraphs for forests of tight paths

In this paper, we investigate the maximal size of a k-uniform hypergraph containing no forests of tight paths, which extends the classical Erdős–Gallai Theorem for paths in graphs. Our results build on the results of Györi, Katona and Lemons, who considered the maximal size of a k-uniform hypergraph containing no single tight path.

A minimum-change version of the Chung–Feller theorem for Dyck paths

A Dyck path of length 2k with e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗-steps and k many ↘-steps that change the current coordinate by (1,1) or (1,−1), respectively, and that has exactly e many ↘-steps below the line y=0. Denoting by D2ke the set of Dyck paths of length 2k with e flaws, the well-known Chung–Feller theorem asserts that the sets D2k0,D2k1,…,D2kk all have the same cardinality 1k+12kk=Ck, the kth Catalan number. The standard combinatorial proof of this classical result establishes a bijection f′ between D2ke and D2ke+1 that swaps certain parts of the given Dyck path x, with the effect that x and f′(x) may differ in many positions. In this paper we strengthen the Chung–Feller theorem by presenting a simple bijection f between D2ke and D2ke+1 which has the additional feature that x and f(x) differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of f in constant time per generated Dyck path. As an application, we use our minimum-change bijection f to construct cycle-factors in the odd graph O2k+1 and the middle levels graph M2k+1 – two intensively studied families of vertex-transitive graphs – that consist of Ck many cycles of the same length.

A minimum-change version of the Chung-Feller theorem for Dyck paths

A Dyck path of length 2k with e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗-steps and k many ↘-steps that change the current coordinate by (1, 1) or (1, −1), respectively, and that has exactly e many ↘-steps below the line y=0. Denoting by D2ke the set of Dyck paths of length 2k with e flaws, the well-known Chung-Feller theorem asserts that the sets D2k0,D2k1,…,D2kk all have the same cardinality 1k+1(2kk)=Ck, the k-th Catalan number. The standard combinatorial proof of this classical result establishes a bijection f′ between D2ke and D2ke+1 that swaps certain parts of the given Dyck path x, with the effect that x and f′(x) may differ in many positions. In this work we strengthen the Chung-Feller theorem by presenting a simple bijection f between D2ke and D2ke+1 which has the additional feature that x and f(x) differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of f in constant time per generated Dyck path. As an application, we use our minimum-change bijection f to construct cycle-factors in the odd graph O2k+1 and the middle levels graph M2k+1 — two intensively studied families of vertex-transitive graphs — that consist of Ck many cycles of the same length.

Augmenting trail theorem for the maximum 1-2 matching problem

We consider the maximum 1-2 matching problem in bipartite graphs. The notion of the augmenting trail for the 1-2 matching problem, which is the extension of the notion of the augmenting path for the 1-1 matching problem is introduced. The main purpose of the present paper is to prove “the augmenting trail theorem” for the 1-2 matching problem in the bipartite graph, which is an analogue of the augmenting path theorem by Bergé for the usual 1-1 matching problems.

Extension theorem for rough paths via fractional calculus

On the basis of fractional calculus, we introduce an integral of weakly controlled paths, which is a generalization of integrals in the context of rough path analysis. As an application, we provide an alternative proof of Lyons' extension theorem for geometric Hölder rough paths together with an explicit expression of the extension map.

An automatic scaling method for obtaining the trace and parameters from oblique ionogram based on hybrid genetic algorithm

AbstractScaling oblique ionogram plays an important role in obtaining ionospheric structure at the midpoint of oblique sounding path. The paper proposed an automatic scaling method to extract the trace and parameters of oblique ionogram based on hybrid genetic algorithm (HGA). The extracted 10 parameters come from F2 layer and Es layer, such as maximum observation frequency, critical frequency, and virtual height. The method adopts quasi‐parabolic (QP) model to describe F2 layer's electron density profile that is used to synthesize trace. And it utilizes secant theorem, Martyn's equivalent path theorem, image processing technology, and echoes' characteristics to determine seven parameters' best fit values, and three parameter's initial values in QP model to set up their searching spaces which are the needed input data of HGA. Then HGA searches the three parameters' best fit values from their searching spaces based on the fitness between the synthesized trace and the real trace. In order to verify the performance of the method, 240 oblique ionograms are scaled and their results are compared with manual scaling results and the inversion results of the corresponding vertical ionograms. The comparison results show that the scaling results are accurate or at least adequate 60–90% of the time.

The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in Long and Zhu (Ann Math 155:317–368, 2002), and get an enhanced version of it. Then we give several applications providing lower bounds on the number of distinct closed geodesics in various settings on compact simply-connected Finsler manifolds.

Maxwell and Rankine reciprocal diagrams via Minkowski sums for two-dimensional and three-dimensional trusses under load

A method is presented for computing and displaying the Maxwell (two-dimensional) or Rankine (three-dimensional) diagram reciprocal to a truss under load. Reciprocals are constructed via the two-dimensional Airy stress function or its three-dimensional analogue. A linear combination of the coordinates of each original node and the coordinates of its reciprocal object leads to the finished diagram. This is related to an underlying Minkowski sum of the polyhedral stress functions of the original and reciprocal diagrams. Some regions of the resulting combined diagram have units of length [Formula: see text] and other regions use units of force [Formula: see text]These regions are connected in a consistent manner, in two dimensions by rectangular areas and in three dimensions by prismatic volumes, each having work units of dimension [Formula: see text] and which give a visual representation of the [Formula: see text] terms in Maxwell’s load path theorem. A linear combination weighted in favour of the original leads to a diagram where original bars are augmented by a thickness (in two-dimensional) or a polygonal cross-sectional area (in three-dimensional). The diagrams thus give an appealing visual representation of the material required for a constant stress design. Although the method gives a complete description of loaded trusses in two dimensions, in three dimensions, it is currently only applicable to those trusses which possess a Rankine reciprocal.

A procedure for the reliability improvement of the oblique ionograms automatic scaling algorithm

AbstractA procedure made by the combined use of the Oblique Ionogram Automatic Scaling Algorithm (OIASA) and Autoscala program is presented. Using Martyn's equivalent path theorem, 384 oblique soundings from a high‐quality data set have been converted into vertical ionograms and analyzed by Autoscala program. The ionograms pertain to the radio link between Curtin W.A. (CUR) and Alice Springs N.T. (MTE), Australia, geographical coordinates (17.60°S; 123.82°E) and (23.52°S; 133.68°E), respectively. The critical frequency foF2 values extracted from the converted vertical ionograms by Autoscala were then compared with the foF2 values derived from the maximum usable frequencies (MUFs) provided by OIASA. A quality factor Q for the MUF values autoscaled by OIASA has been identified. Q represents the difference between the foF2 value scaled by Autoscala from the converted vertical ionogram and the foF2 value obtained applying the secant law to the MUF provided by OIASA. Using the receiver operating characteristic curve, an appropriate threshold level Qt was chosen for Q to improve the performance of OIASA.

Footwear-related variability in running

The manner of adapting one's running biomechanics when wearing different footwear designs appears to be individual-specific. Applying the preferred movement path theorem, wearing less than optimal running shoes bears the risk of forcing the joints to leave their preferred path and load unadapted joint areas that might be overused during sport activities. The goal of this article is to introduce a method quantifying a person's biomechanical responsiveness towards footwear, the so-called ‘footwear-related variability’, and to explore corresponding determinants. Eighty-nine participants were tested wearing six different running shoe conditions. They performed isometric strength tests for knee and ankle extensor muscles. Further, anthropometrics, as well as lower extremity kinematics and kinetics during linear running at 3.5 m/s were recorded. Their stance phase normalized waveform patterns of running biomechanical data were analysed by means of the ‘adjusted coefficient of multiple determination’, quantifying how much footwear-induced biomechanical response occurred. The results showed that knee joint-specific footwear-related variability was higher in female participants than in male participants. Strength-generating capacity correlated with some footwear-related variability measures and men were found to be stronger than women. Footwear-related variability appeared to be influenced by the extent of knee adduction, hip inward rotation and ankle outward rotation moment, which are characteristic to distinguish between men's and women's running styles. Anatomical and anthropometric differences between men and women as well as strength capacity disadvantages might make women more susceptible to footwear-induced running biomechanical alterations. Future studies should investigate the relationship between footwear-related variability and overuse injuries, in order to find critical thresholds of footwear-related variability and identify groups of runners that need to be specifically targeted with injury preventing measures.

Refinements of [formula omitted]-Dyck paths

The classical Chung–Feller theorem tells us that the number of ( n , m ) -Dyck paths is the n th Catalan number and independent of m . In this paper, we consider refinements of ( n , m ) -Dyck paths by using four parameters, namely the peak, valley, double descent and double ascent. Let p n , m , k be the total number of ( n , m ) -Dyck paths with k peaks. First, we derive the reciprocity theorem for the polynomial P n , m ( x ) = ∑ k = 1 n p n , m , k x k . In particular, we prove that the number of ( n , m ) -Dyck paths with k peaks is equal to the number of ( n , n − m ) -Dyck paths with n − k peaks. Then we find the Chung–Feller properties for the sum of p n , m , k and p n , m , n − k , i.e., the number of ( n , m ) -Dyck paths which have k or n − k peaks is 2 ( n + 2 ) n ( n − 1 ) n k − 1 n k + 1 for 1 ≤ m ≤ n − 1 and independent of m . Finally, we provide a Chung–Feller type theorem for Dyck paths of semilength n with k double ascents: the total number of ( n , m ) -Dyck paths with k double ascents is equal to the total number of n -Dyck paths that have k double ascents and never pass below the x -axis, which is counted by the Narayana number. Let v n , m , k (resp. d n , m , k ) be the total number of ( n , m ) -Dyck paths with k valleys (resp. double descents). Some similar results are derived.

A visible shoreline theorem for staircase paths

Let C be an orthogonal polygon in the plane, bounded by a simple closed curve, and assume that C is starshaped via staircase paths. Let \(P \subseteq {\mathbb{R}}^2\backslash({\rm int} C)\). If every four points of P see a common boundary point of C via staircase paths in \({\mathbb{R}}^2\backslash({\rm int} C)\), then there is a boundary point b of C such that every point of P sees b (via staircase paths in \({\mathbb{R}}^2\backslash({\rm int} C)\)). The number four is best possible, even if C is orthogonally convex.

Chebyshev type lattice path weight polynomials by a constant term method

We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary ‘decorated’ weights as well as an arbitrary ‘background’ weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence, we also present an efficient method for finding explicit closed-form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed-form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as the models of steric stabilization and sensitized flocculation.

Stability of the path–path Ramsey number

Here we prove a stability version of a Ramsey-type Theorem for paths. Thus in any 2-coloring of the edges of the complete graph K n we can either find a monochromatic path substantially longer than 2 n / 3 , or the coloring is close to the extremal coloring.