Polynomial Argument Research Articles

Automated Train Coordinate Determination System with Self-Tuning of the Decision Function

Introduction. The problem of determining the train coordinates on the approach section to the crossing is associated with the impact of destabilizing factors on the information primary detector ? the rail line with distributed parameters. This leads to an error in calculating train coordinates. The aim of the study is to develop and scientifically substantiate the principle of building a system for calculating train coordinates with self-tuning of the decision function under the influence of significant destabilizing factors on the information primary sensor. Materials and Methods. To solve the problem of reliable determination of train coordinates, we propose a two-phase principle for forming the decision function. At the first stage, by means of a training sample of images and using the learning principle, the decision function (model) of the system for calculating train coordinates is determined. When the train enters a fixed-length approach section, the mismatch is determined by comparing the calculated coordinate with the fixed one. The second stage is the self-tuning of the coefficients of the decision function until the required accuracy is achieved. Results. The article shows the stages of forming the decision function by two-dimensional images; there was developed and tested an algorithm for self-turning of the decision function under the influence of various destabilizing factors. Through using 6 attributes of components of current and voltage vectors at the rail line input, 6 solving functions were obtained. Various combinations of two-dimensional images were used as polynomial arguments. Discussion and Conclusion. The study results confirm the feasibility of forming decision function and its self-tuning. The maximum error in calculating coordinates for various combinations ranges from 9.97% (199.34 m) to 4.57% (91.49 m). The error of determination of 5% for two decisive functions satisfies the safety requirements, since in a 45-second time interval to activate an automatic crossing signal, a distance of 100 m is covered in 3 seconds, i.e. the elapsed time is only 3 seconds in a 45 second interval.

On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).

Density modulo 1 of a sequence associated to some multiplicative functions evaluated at polynomial arguments

We study density modulo 1 of the sequence with general term ∑ m≤n f (G(m)) where f is the strongly multiplicative function of the form f (n) = ∏ p|n (1 − ν(p) p) and ν is a multiplicative function for which there exists a real number 0 < r ≤ 1 such that 1 ≤ |ν(p)| ≤ p

Trace formulas for general Hermitian matrices: unitary scattering approach and periodic orbits on an associated graph

Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. In the first type the unitary matrix is where is the spectral parameter. The new feature is that the spectral parameter appears in the final form as an argument of Eulerian polynomials—thus connecting the periodic orbits to combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is that the locations and radii of the spectral discs of Gershgorin’s theorem appear naturally as the pole parameters of the scattering matrix. Both formulas are discussed and possible applications are outlined.

A Partial Theta Function Borwein Conjecture

We present an infinite family of Borwein type + - - conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.

New identities for terminating balanced $$_4\phi _3$$-series

The duplicate form of the Carlitz inversions is employed to prove new identities for terminating balanced $$_4\phi _3$$-series. Their limiting series are deduced as consequences. The “dual formulae” are also derived by means of the polynomial argument.

Some New q-Congruences for Truncated Basic Hypergeometric Series

We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews’ multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.

Polynomial argument for $q$-binomial cubic sums*

By means of the polynomial argument, a class of cubic sums of $q$-binomial coefficients are evaluated in closed forms.

Three theorems on odd degree Chebyshev polynomials and more generalized permutation polynomials over a ring of module $$2^w$$ 2 w

Odd degree Chebyshev polynomials over a ring of modulo $$2^w$$ have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterates affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observed when changing the degree of Chebyshev polynomials with a fixed argument of polynomials. Both kinds of period have not been completely studied. In this paper, we clarify completely both of them. The knowledge about them enables us to efficiently solve degree decision problem of Chebyshev polynomial over the ring, and so a key-exchange protocol with Chebyshev polynomial over the ring is not secure. In addition, we generalize the discussion and show that a key-exchange protocol with more generalized permutation polynomials which belong to a certain class is not secure.

Optimal Learning Rates for Kernel Partial Least Squares

We study two learning algorithms generated by kernel partial least squares (KPLS) and kernel minimal residual (KMR) methods. In these algorithms, regularization against overfitting is obtained by early stopping, which makes stopping rules crucial to their learning capabilities. We propose a stopping rule for determining the number of iterations based on cross-validation, without assuming a priori knowledge of the underlying probability measure, and show that optimal learning rates can be achieved. Our novel analysis consists of a nice bound for the number of iterations in a priori knowledge-based stopping rule for KMR and a stepping stone from KMR to KPLS. Technical tools include a recently developed integral operator approach based on a second order decomposition of inverse operators and an orthogonal polynomial argument.

Summation formulae for a class of terminating balanced q-series

A class of terminating balanced q-series are investigated. Thirty summation formulae are established by employing Carlitz' inversions and the polynomial argument, as well as contiguous relations.

Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.

Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem

We prove the discrete analogue of Kakeya conjecture over $$\mathbb {R}^n$$ . This result suggests that a (hypothetically) low-dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all dimensions. The main tool of the proof is a theorem of Wongkew (Pac J Math 159:177–184, 2003), which states that a low-degree polynomial cannot have its zero set being too dense inside the unit cube, coupled with Dvir-type polynomial arguments (Dvir in J Am Math Soc 22(4):1093–1097, 2009). From the viewpoint of the proofs, we also state a conjecture that is stronger than and almost equivalent to the (lower) Minkowski version of the Kakeya conjecture and prove some results toward it. We also present our own version of the proof of the theorem in Wongkew (Pac J Math 159:177–184, 2003). Our proof shows that this theorem follows from a combination of properties of zero sets of polynomials and a general proposition about hypersurfaces which might be of independent interest. Finally, we discuss how to generalize Bourgain’s conjecture to high dimensions, along the way providing a counterexample to the most naive generalization.

Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA).

On pseudorandom sequences of $$k$$ k symbols constructed using finite fields

A family of pseudorandom sequences of $$k$$ k symbols are constructed by using finite fields of prime-power order. The construction is an extension of certain construction of Sarkozy and Winterhof on binary sequences using the quadratic character with polynomial arguments over any finite fields, and of certain construction of Ahlswede, Mauduit and Sarkozy on sequences of $$k$$ k symbols using multiplicative characters with polynomial arguments over finite prime fields. Certain pseudorandom measures of the resulting sequences are considered.

On the distribution of a linear sequence associated to sum of divisors evaluated at polynomial arguments

Following the recent method of Deshouillers and the author in the theory of distribution modulo 1, we show that the sequence with general term bn = ?m?n(m2 + 1)/?(m2 + 1) is dense modulo 1.

Branching rules for symmetric functions and [formula omitted] basic hypergeometric series

A one-parameter rational function generalisation Rλ(X;b) of the symmetric Macdonald polynomial and interpolation Macdonald polynomial is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for Rλ(X;b). Our main motivation for studying Rλ(X;b) is that it leads to a new class of sln basic hypergeometric series, generalising the well-known basic hypergeometric series with Macdonald polynomial argument. For these new series we prove sln analogues of the q-Gauss and q-Kummer–Thomae–Whipple formulas. In a special limit, one of our results implies an elegant binomial formula for Jack polynomials, different to that of Kaneko, Lassalle, Okounkov and Olshanski.

Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

In two previous papers, the first named author jointly with Florian Luca and Henryk Iwaniec, have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers. In this note, we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values, proving the density modulo 1 of the sequence $$ \left( {\sum\limits_{m \leqslant n} {\frac{{\phi (m^2 + 1)}} {{m^2 + 1}}} } \right)_n . $$ This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.

Measures of pseudorandomness for binary sequences constructed using finite fields

We extend the results of Goubin, Mauduit and Sárközy on the well-distribution measure and the correlation measure of order k of the sequence of Legendre sequences with polynomial argument in several ways. We analyze sequences of quadratic characters of finite fields of prime power order and consider in each case two, in general, different definitions of well-distribution measure and correlation measure of order k , respectively.

A new multivariable 6 ψ 6 summation formula

By multidimensional matrix inversion, combined with an A_r extension of Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7 summation is derived. By a polynomial argument this 8-phi-7 summation is transformed to another multivariable 8-phi-7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6-phi-5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey's very-well-poised 6-psi-6 summation formula.