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The \begin{document}$ k $\end{document} -deck of a sequence is defined as the multiset of all its subsequences of length \begin{document}$ k $\end{document} . Let \begin{document}$ D_k(n) $\end{document} denote the number of distinct \begin{document}$ k $\end{document} -decks for binary sequences of length \begin{document}$ n $\end{document} . For binary alphabet, we determine the exact value of \begin{document}$ D_k(n) $\end{document} for small values of \begin{document}$ k $\end{document} and \begin{document}$ n $\end{document} , and provide asymptotic estimates of \begin{document}$ D_k(n) $\end{document} when \begin{document}$ k $\end{document} is fixed. Specifically, for fixed \begin{document}$ k $\end{document} , we introduce a trellis-based method to compute \begin{document}$ D_k(n) $\end{document} in time polynomial in \begin{document}$ n $\end{document} . We then compute \begin{document}$ D_k(n) $\end{document} for \begin{document}$ k \in \{3,4,5,6\} $\end{document} and \begin{document}$ k \leqslant n \leqslant 30 $\end{document} . We also improve the asymptotic upper bound on \begin{document}$ D_k(n) $\end{document} , and provide a lower bound thereupon. In particular, for binary alphabet, we show that \begin{document}$ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $\end{document} and \begin{document}$ D_k(n) = \Omega(n^k) $\end{document} . For \begin{document}$ k = 3 $\end{document} , we moreover show that \begin{document}$ D_3(n) = \Omega(n^6) $\end{document} while the upper bound on \begin{document}$ D_3(n) $\end{document} is \begin{document}$ O(n^9) $\end{document} .

An interaction function was constructed based on the axial compression/tension and shear loads of orthotropic plates. The coefficients of the polynomial function were determined by uniaxial test results. Buckling interaction and failure interaction formulae under combined axial tension/compression and shear loads were established. Based on the uniaxial load test results of orthotropic plates, the buckling load and bearing capacity under any proportion of the combined loads could be predicted by using the proposed interaction formulae. The buckling interaction curves and failure envelopes predicted by the proposed interaction formulae were in excellent agreement with the test results.

We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.

A float detects the level of a liquid in a tank or a container. It floats on top of the liquid surface and acts as a mechanical switch when the water moves up or down in the auto drain valve tank. The auto drain valve removes the water in the compressed air circuit The spherical float used in the auto drain valve is prone to the realization of a dent and the float joint fails due to the high pressure of air in the compressed air circuit. In this paper, the spherical float was analysed numerically to select the suitable stainless steel materials out of SS 302, SS 305, SS 312, SS 316 to withstand the high-pressure force and temperature of the air in the drain valve. The applied pressure was varied from 0.7 to 2.0 MPa and the temperature from 2 °C to 80 °C based on the drain valve application. The structural analysis results obtained were the total deformation, the equivalent stress, and the pressure at the joints of the float. Equations were obtained to get the relationships between applied pressure and temperature with equivalent stress and total deformation. The results indicate that both equivalent stress and total deformation varies linearly with pressure. With respect to temperature, equivalent stress varies linearly and total deformation varies in polynomial relation. The minimum and maximum values of total deformation obtained were 0 to 0.0132 mm, equivalent stress were 37.11 to 75.40 MPa and the average joint pressure was 12.43 to 76.56 MPa. The results show that SS 314 grade could be chosen for fabrication of spherical float in drain valve application.

Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{\mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){B_N/SU(N)}. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description ofBN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.

The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all $$R^{K}_{{\varvec{0}}}$$ -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an $$ER^{K}$$ -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature.

The objective was to develop a visco-elastic-plastic model for the prediction and characterisation of the compaction behaviour of sugarcane agricultural residues during the production of bales under different moisture contents (27%, 37% and 48%) and different particle sizes (∼300 mm, 150 mm and 80 mm). Data of confined-compaction and stress relaxation experiments were used to calibrate the model parameters. The proposed constitutive equation was composed of a generalised Maxwell model with three branches in series with a plastic component. The procedure was developed to characterise model parameters and used a numeric approach to calculate the hereditary integral when fitting the experimental curves, whilst the stress due to the plastic strains was fitted with a polynomial. The model provided good accuracy (R2 > 0.92) to fit the compaction and relaxation stages with the same set of parameters. Thus, it predicted the energy required to produce bales as well as the final relaxed density of this sustainable material that may become an important source of energy.

The aim of the paper is to investigate all transcendental entire solutions of the following general nonlinear equation of the form $f^{n}+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ , where Pd(z,f) is a differential-difference polynomial in f of degree d. Our result is a generalization and complement of known results obtained by Liu-Mao, L ${\ddot {\mathrm {u}}}$ et al. and the references therein.

Let A be a Poisson algebra over a field $$\mathbf{k}$$ with characteristic zero, let $$\gamma $$ , $$\alpha $$ be Poisson derivations on A such that $$\gamma \alpha =\alpha \gamma $$ and $$0\ne \rho \in \mathbf{k}$$ . Here the notion of a $$\gamma $$ -Poisson normal element is introduced, it is proved that the polynomial algebra A[y, x] has a Poisson structure defined by $$\{y,a\}=\alpha (a)y, \{x,a\}=\beta (a)x, \{x,y\}=\beta (y)x+\delta (y)$$ for $$a\in A$$ , where $$\beta $$ is a Poisson derivation on A[y] defined by $$\beta |_A=\gamma -\alpha $$ , $$\beta (y)=\rho y$$ and $$\delta $$ is a derivation on A[y] such that $$\delta |_A=0$$ , and its Poisson simplicity criterion is established and endorsed by examples.

In this paper, for the complex Lie algebra sl3 we propose an explicit formula for finding the multiplicity of the weight of the irreducible representation Γλ, which is determined by its higher weight λ = (a,b). The set of all weights Λ of such a representation forms a group ring Z[Λ] with the multiplicative basis e(μ),μ ∈ Λ The character of the representation Char Γλ is an element of Z[Λ], the coefficients of which are the required multiplicities.The main idea of the calculations is to specify the basis e(μ) = xμ1yμ2 of the group ring Z[Λ]. This made it possible to represent the character Char Γλ of the irreducible representation Γλ as a Schur polynomial $ s_ {a, b} \left (x, \dfrac { y} {x}, \dfrac {1} {y} \right) $ of two variables $ x, y $. As a consequence, we express the coefficients of this polynomial through simple functions that are easily computed for linear time. The key role in the calculation was played by the explicitly found coefficients of the series decomposition$$\Delta = \dfrac {1} {\left ({y} ^ {2} -x \right) \left (1- yx \right) \left (y- {x} ^ {2} \right)},$$in terms of the function $$c (n, k) = \left \{\begin {array} {l}\min (n {-} k + 2, k), 1 \leq k \leq n + 1, \\\\0, \text {{\rm otherwise.}}\end {array}\right.$$

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the $\mathbb {Z}_{2} \mathbb {Z}_{4}$ -additive quasi-cyclic codes.

In this paper, a functionally graded material column that is simply supported at one end and clamped at the other is considered. The buckling mode is postulated as a high-order polynomial. Six nove...

This paper proposes the Guruswami-Sudan (GS) list decoding algorithm for one-point elliptic codes, in which the interpolation is realized by the module basis reduction (BR). Elliptic codes are a kind of algebraic-geometric (AG) codes with a genus of one. Over the same finite field, they have a greater codeword length than Reed-Solomon (RS) codes, capable of correcting more errors. The GS decoding consists of interpolation and root-finding, while the former that determines the interpolation polynomial $\mathcal {Q}(\text {x}, \text {y}, \text {z})$ dominates the decoding complexity. By defining the Lagrange interpolation function over an elliptic function field, a basis of the interpolation module can be constructed. The desired Grobner basis that contains $\mathcal {Q}(\text {x}, \text {y}, \text {z})$ can be determined by reducing the constructed basis. This is namely the BR interpolation and it requires less finite field arithmetic operations than the conventional Kotter’s interpolation, facilitating the GS decoding. Re-encoding transform (ReT) is further introduced to facilitate the BR interpolation. This work also shows that both the BR interpolation and its ReT variant will have a lower complexity as the code rate ${k}/{n}$ increases, where n and ${k}$ are the length and dimension of the code, respectively. Our numerical results demonstrate the complexity advantage of the BR interpolation over Kotter’s interpolation, and the performance advantage of elliptic codes over RS codes.

Let $P: {\mathbb C} \to {\mathbb C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is non-degenerate, then $z_0$ is the landing point of at least one {\it smooth} external ray. The statement is optimal in the sense that all but one ray landing at $z_0$ may be broken., Comment: 10 pages, 3 figures

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.

A generalization of the theory of Y. Brudnyi \cite{yuri}, and A. and Y. Brudnyi \cite{BB20a}, \cite{BB20b}, is presented. Our construction connects Brudnyi's theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak--Kuznetsov. Our spaces shed light on the structure of the John--Nirenberg spaces. We show that $SJN_{p}$ (sparse John--Nirenberg space) coincides with $L^{p},1

It is known that polarizationless P systems with active membranes can solve $$\mathrm {PSPACE}$$ PSPACE -complete problems in polynomial time without using in-communication rules but using the classical (also called strong) non-elementary membrane division rules. In this paper, we show that this holds also when in-communication rules are allowed but strong non-elementary division rules are replaced with weak non-elementary division rules, a type of rule which is an extension of elementary membrane divisions to non-elementary membranes. Since it is known that without in-communication rules, these P systems can solve in polynomial time only problems in $$\mathrm {P}^{\text {NP}}$$ P NP , our result proves that these rules serve as a borderline between $$\mathrm {P}^{\text {NP}}$$ P NP and $$\mathrm {PSPACE}$$ PSPACE concerning the computational power of these P systems.

Given a complex polynomial p(z) of degree n, we are interested in characterization and computation of local optima — maxima or minima — of |p| over a disc D of radius r centered at the origin, or its boundary, ∂D. From the maximum modulus principle (MMP), together with parametrization of ∂D, maximization reduces to a univariate optimization. However, reduction to univariate optimization does not guarantee efficiency in the computation of local optima, nor gives any specific characterization, nor provides a bound on the number of local optima. In this article, we utilize the geometric modulus principle, a stronger result than MMP, to prove novel characterizations of local optima over D or ∂D. On the one hand, we prove any local optimum that lies on ∂D is necessarily a fixed point of $F_{r}(z)=\pm r(p/p^{\prime })/|p/p^{\prime }|$ and also give a sufficiency condition. On the other hand, we show any fixed point is necessarily a root of a derived polynomialPr(z) of degree 2n defined in terms of p, $p^{\prime }$ and their conjugate reciprocals. This also proves the number of local optima is at most 2n, an upper bound that can be attained. Our results imply that all desired local optima can be approximated to arbitrary precision in polynomial time. In particular, the classic $\| p \|_{\infty }$ can be computed efficiently. Implementation of our algorithm using Matlab and MPSolve is capable of computing all roots of Pr(z) of degree up to 250 and subsequently all local optima of |p| to precision of 10− 13 in less than 3 s. In summary, the proposed theoretical algorithm is also practical and efficient. Additionally, based on the derived formulas we propose specialized algorithms and present sample visualization through polynomiography.

"In this paper, we introduce new class $\Sigma_{m}(\mu,\lambda,\gamma,\beta)$ of $m$-fold symmetric bi-univalent functions. Furthermore, we use the Faber polynomial expansions to find upper bounds for the general coefficients $|a_{mk+1}|(k \geqq 2)$ of functions in the class $\Sigma_{m}(\mu,\lambda,\gamma,\beta)$. Moreover, we obtain estimates for the initial coefficients $|a_{m+1}| $ and $|a_{2m+1}|$ for functions in this class. The results presented in this paper would generalize and improve some recent works."

This paper gives explicit constructions of all irreducible representations of unipotent radicals $$N_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ of the standard parabolic subgroups $$P_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ of $${\mathrm {GL}}_n({\mathbb {F}}_q),$$ corresponding to the ordered partition $$ (n_1,\ n_2,\ n_3)$$ of n. The construction gives a bijection between coadjoint orbits of $$N_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ and irreducible representations inducing from degree 1 characters in the sense of Boyarchenko’s construction. The result shows that the number of irreducible characters of $$N_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ with a fixed degree is a polynomial in $$q-1$$ with nonnegative integer coefficients and verifies analogue conjectures of Higman, Lehrer, and Isaacs.

Experimental data of density have been measured for binary liquid mixtures containing {amines (n-butylamine, or s-butylamine, or t-butylamine, or diethylamine, or propylamine) + water}, over the entire range of composition at temperatures between 283.15 and 303.15 K, and atmospheric pressure. The density values enabled the determination of the thermal expansion coefficients. The excess molar volume, $${\text{V}}_{\text{m}}^{\text{E}}$$ , was calculated using the experimental data, from which a Redlich–Kister type polynomial was fit, enabling the determination of the partial molar volumes, the excess partial molar volumes, the apparent molar volumes and the excess partial molar volumes at infinite dilution. The $${\text{V}}_{\text{m}}^{\text{E}}$$ values were also used to test the applicability of the Extended Real Associated Solution Model (ERAS Model). The results for the studied systems suggest that structural effects and chemical interactions must predominate over other possible effects. The magnitude of $${\text{V}}_{\text{m}}^{\text{E}}$$ for the studies systems led to the following order: n-butylamine > propylamine > s-butylamine > diethylamine > t-butylamine.

Adoption of heavy haul traffic on many railroads, comprising Russian railways, has highlighted the relevance of assessing the effect of increased axial loads on the contact fatigue life of rails.The article describes a set of theoretical studies carried out to create a scientifically substantiated method for predicting the contact fatigue life of rails depending on the values of axial loads. The stress-strain state of the contact area has been determined using the finite element model of wheel rolling on a rail. It has been found that the wheel-rail rolling contact area undergoes complex multiaxial loading with the simultaneous action of normal and shear strains. Based on the analysis of models describing multiaxial fatigue damage, the Brown–Miller model was chosen, which considers the simultaneous action of normal strains at the contact area and of maximum shear strains, which most fully describes the stress-strain state of the wheel-rail rolling contact area. To apply the Brown–Miller model, fatigue stress-strain curves for rail steel have been identified. Based on the analysis of methods for determining the parameters of stress-strain curves carried out by V. A. Troschenko, a modified Roessle– Fatemi hardness method has been applied. Based on the experimentally determined values of hardness on the rolling surface, the parameters of the curves of elastic and plastic fatigue have been revealed by calculation and experiment. To establish the damaging effect of the load from wheel rolling on a rail, the concept of relative damage per rolling cycle had been assumed which is the value inverse to the number of cycles preceding formation of a contact-fatigue crack at a given value of the axial load.Calculations of the relative damage rate of the rolling surface of rails caused by contact fatigue defects were carried out with the Fatigue software package considering mean values of the indicators of the degree of fatigue strength and plasticity of rail steel and the calculated stresses in the wheel-rail contact area, as well as the plasticity correction using Neuber method. The polynomial dependence of relative damageability of the rolling surface of rails is obtained. The established functional dependence of relative damageability of the rolling surface of rails on the values of vertical forces can be used as the basis for the developed methodology for predicting the contact fatigue life of rails.

We provide a graphical treatment of SAT and #SAT on equal footing. Instances of #SAT can be represented as tensor networks in a standard way. These tensor networks are interpreted by diagrams of the ZH-calculus: a system to reason about tensors over C in terms of diagrams built from simple generators, in which computation may be carried out by transformations of diagrams alone. In general, nodes of ZH diagrams take parameters over C which determine the tensor coefficients; for the standard representation of #SAT instances, the coefficients take the value 0 or 1. Then, by choosing the coefficients of a diagram to range over B, we represent the corresponding instance of SAT. Thus, by interpreting a diagram either over the boolean semiring or the complex numbers, we instantiate either the decision or counting version of the problem. We find that for classes known to be in P, such as 2SAT and #XORSAT, the existence of appropriate rewrite rules allows for efficient simplification of the diagram, producing the solution in polynomial time. In contrast, for classes known to be NP-complete, such as 3SAT, or #P-complete, such as #2SAT, the corresponding rewrite rules introduce hyperedges to the diagrams, in numbers which are not easily bounded above by a polynomial. This diagrammatic approach unifies the diagnosis of the complexity of CSPs and #CSPs and shows promise in aiding tensor network contraction-based algorithms., Comment: In Proceedings QPL 2020, arXiv:2109.01534

We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of a polynomial growth Cayley graph $\Gamma$. We prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to $1/|B|^{\frac{1}{d-1}}$, where $d$ represents the growth rate of $\Gamma$. The method consists in associating a manifold $M$ to $\Gamma$ and a bounded domain $N \subset M$ to a subgraph $(\Omega, B)$ of $\Gamma$. We find upper bounds for the Steklov spectrum of $N$ and transfer these bounds to $(\Omega, B)$ by discretizing $N$ and using comparison Theorems., Comment: 17 pages

In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in $$S_n$$ , the set of all permutations on n elements, under the Kendall $$\tau $$ -metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in $$S_n$$ , under the Kendall $$\tau $$ -metric, for more values of n. Specifically, we present the polynomial representation of the size of a ball in $$S_n$$ under the Kendall $$\tau $$ -metric for some radius r, and obtain some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect t-error-correcting code in $$S_n$$ under the Kendall $$\tau $$ -metric for some n and $$t=2,3,4,5,~\text {or}~\frac{5}{8}\left( {\begin{array}{c}n\\ 2\end{array}}\right) < 2t+1\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ .

Let a set of nodes $$\mathcal{X}$$ in the plane be $$n$$ -independent, i.e., each node has a fundamental polynomial of degree $$n.$$ Assume that $$\#\mathcal{X}=d(n,k-3)+3=(n+1)+n+\cdots+(n-k+5)+3$$ and $$4\leq k\leq n-1.$$ In this paper we prove that there are at most seven linearly independent curves of degree less than or equal to $$k$$ that pass through all the nodes of $$\mathcal{X}.$$ We provide a characterization of the case when there are exactly seven such curves. Namely, we prove that then the set $$\mathcal{X}$$ has a very special construction: all its nodes but three belong to a (maximal) curve of degree $$k-3.$$ Let us mention that in a series of such results this is the third one. At the end, an important application to the bivariate polynomial interpolation is provided, which is essential also for the study of the Gasca–Maeztu conjecture.

Two examples illustrating properties of discrete measures are given. In the first part of the paper, it is proved that, for any probability measure $$\mu$$ with $$\operatorname{supp}{\mu}=[-1,1]$$ whose logarithmic potential is continuous on $$[-1,1]$$ , there exists a (discrete) measure $$\sigma=\sigma(\mu)$$ with $$\operatorname{supp}{\sigma}=[-1,1]$$ such that the corresponding orthogonal polynomials $$P_n(x;\sigma)=x^n+\dotsb$$ satisfy the condition $$(1/n)\chi(P_n(\,\cdot\,;\sigma))\xrightarrow{*}\mu$$ , $$n\to\infty$$ , where $$\chi(\,\cdot\,)$$ is the measure counting the zeros of a polynomial. The proof of the existence of such a measure $$\sigma$$ is based on properties of weighted Leja points. In the second part, an example of a compact set and a sequence of discrete measures supported on it with a special property is given. Namely, the sequence of measures converges in the $$*$$ -weak topology to the equilibrium measure on the compact set, but the corresponding sequence of logarithmic potentials converges in capacity to the equilibrium potential in no neighborhood of this compact set.

Wave propagation in a system of two arbitrary fluids separated by a contaminated interface in the form of an insoluble monolayer is investigated. The effects of four parameters (density ratio, R ; kinematic viscosity ratio, V ; Marangoni number, P ; and squared wave-Reynolds number, σ ) on the solutions of a comprehensive dispersion relation for these waves are analyzed. In the first part of the manuscript, several special cases are considered wherein the comprehensive dispersion relation is modified to a simple polynomial equation and its numerical solutions are obtained for σ ∈ [ 0 , 1 ] . A bifurcation is observed at σ = σ b in two of the solutions in each case; and the system is shown to be overdamped for σ σ b , and underdamped for σ > σ b . The relative influences of { R , V , P } on σ b are also analyzed, and it is found that in general, σ b is a strong function of { R , V } , but a weak function of P . In the second part of the manuscript, numerical solutions of the comprehensive dispersion relation are obtained for waves in a decane-water system, in the limit of { P ≫ 1 , σ ≫ 1 } ; and these solutions are classified to correspond to a transverse wave or to a longitudinal wave. The presence of a maximum/minimum in the frequency of oscillation and in the damping rate is observed for the transverse/longitudinal waves. For both the transverse and longitudinal wave modes, the relative deviation of the numerical solution (obtained from the comprehensive dispersion relation) from the respective approximate solution (obtained from a simple dispersion relation) is calculated. The relative deviation, though small, cannot be neglected because it accounts for the effects of elasticity and of capillarity/gravity; and disregarding it in the analyses will fail to yield the inherent maximum/minimum in the frequency of oscillation and in the damping rate for the transverse/longitudinal waves.

We compute the one-loop quantum corrections to the kink energies of the sinh-deformed $\phi^{4}$ and $\varphi^{6}$ models in one space and one time dimensions. These models are constructed from the well-known polynomial $\phi^{4}$ and $\varphi^{6}$ models by a deformation procedure. We also compute the vacuum polarization energy to the non-polynomial function $U(\phi)=\frac{1}{4}(1-\sinh^{2}\phi)^{2}$. This potential approaches the $\phi^{4}$ model in the limit of small values of the scalar function. These energies are extracted from scattering data for fluctuations about the kink solutions. We show that for certain topological sectors with non-equivalent vacua the kink solutions of the sinh-deformed models are destabilized., Comment: 12 pages, 8 figures, 4 tables

In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when $0\leqslant \beta

We perform rigorous runtime analyses for the univariate marginal distribution algorithm (UMDA) and the population-based incremental learning (PBIL) Algorithm on LeadingOnes. For the UMDA, the currently known expected runtime on the function is $${\mathcal {O}}\left( n\lambda \log \lambda +n^2\right)$$ O n λ log λ + n 2 under an offspring population size $$\lambda =\Omega (\log n)$$ λ = Ω ( log n ) and a parent population size $$\mu \le \lambda /(e(1+\delta ))$$ μ ≤ λ / ( e ( 1 + δ ) ) for any constant $$\delta >0$$ δ > 0 (Dang and Lehre, GECCO 2015). There is no lower bound on the expected runtime under the same parameter settings. It also remains unknown whether the algorithm can still optimise the LeadingOnes function within a polynomial runtime when $$\mu \ge \lambda /(e(1+\delta ))$$ μ ≥ λ / ( e ( 1 + δ ) ) . In case of the PBIL, an expected runtime of $${\mathcal {O}}(n^{2+c})$$ O ( n 2 + c ) holds for some constant $$c \in (0,1)$$ c ∈ ( 0 , 1 ) (Wu, Kolonko and Möhring, IEEE TEVC 2017). Despite being a generalisation of the UMDA, this upper bound is significantly asymptotically looser than the upper bound of $${\mathcal {O}}\left( n^2\right)$$ O n 2 of the UMDA for $$\lambda =\Omega (\log n)\cap {\mathcal {O}}\left( n/\log n\right)$$ λ = Ω ( log n ) ∩ O n / log n . Furthermore, the required population size is very large, i.e., $$\lambda =\Omega (n^{1+c})$$ λ = Ω ( n 1 + c ) . Our contributions are then threefold: (1) we show that the UMDA with $$\mu =\Omega (\log n)$$ μ = Ω ( log n ) and $$\lambda \le \mu e^{1-\varepsilon }/(1+\delta )$$ λ ≤ μ e 1 - ε / ( 1 + δ ) for any constants $$\varepsilon \in (0,1)$$ ε ∈ ( 0 , 1 ) and $$0 0 < δ ≤ e 1 - ε - 1 requires an expected runtime of $$e^{\Omega (\mu )}$$ e Ω ( μ ) on LeadingOnes, (2) an upper bound of $${\mathcal {O}}\left( n\lambda \log \lambda +n^2\right)$$ O n λ log λ + n 2 is shown for the PBIL, which improves the current bound $${\mathcal {O}}\left( n^{2+c}\right)$$ O n 2 + c by a significant factor of $$\Theta (n^{c})$$ Θ ( n c ) , and (3) we for the first time consider the two algorithms on the LeadingOnes function in a noisy environment and obtain an expected runtime of $${\mathcal {O}}\left( n^2\right)$$ O n 2 for appropriate parameter settings. Our results emphasise that despite the independence assumption in the probabilistic models, the UMDA and the PBIL with fine-tuned parameter choices can still cope very well with variable interactions.

The topic of this paper are non-self-adjoint second-order differential operators with constant delay generated by $$-y''+q(x)y(x-\tau )$$ where potential q is complex-valued function, $$q\in L^{2}[0,\pi ]$$ . We study inverse problems of these operators for $$\tau \in \left[ \frac{2\pi }{5},\pi \right) $$ . We investigate the inverse spectral problems of recovering operators from their two spectra, firstly under Dirichlet–Dirichlet and second under Dirichlet/Polynomial boundary conditions. We will prove theorem of uniqueness, and we will give procedure for constructing potential. In the first case, for $$\tau \in \left[ \frac{\pi }{2},\pi \right) :$$ we will show that Fourier coefficients of a potential are uniquely0 determined by spectra. In the second case for $$\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) ,$$ we will construct integral equation under potential and we will prove that this integral equation has a unique solution. Also, we will show that other parameters are uniquely determined by spectra.

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given n copies of a mixed state $$\rho \in \mathbb {C}^{d\times d}$$ and the goal is to distinguish (with high probability) whether $$\rho $$ ’s spectrum satisfies some property $${\mathcal {P}}$$ or whether it is at least $$\epsilon $$ -far in $$\ell _1$$ -distance from satisfying $${\mathcal {P}}$$ . This problem was promoted in the survey of Montanaro and de Wolf (A survey of quantum property testing. Technical report, arXiv:1310.2035 , 2013) under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Unlike property testing probability distributions—where one generally hopes for algorithms with sample complexity that is sublinear in the domain size—here the hope is for algorithms with subquadratic copy complexity in the dimension d. This is because the (frequently rediscovered) “empirical Young diagram (EYD) algorithm” (Alicki et al. in J Math Phys 29(5):1158–1162, 1988; Keyl and Werner in Phys Rev A 64(5):052311, 2011; Hayashi and Matsumoto in Phys Rev A 66(2):022311, 2002; Christandl and Mitchison in Commun. Math. Phys. 261(3):789–797, 2006) can estimate the spectrum of any mixed state up to $$\epsilon $$ -accuracy using only $${\widetilde{O}}(d^2/\epsilon ^2)$$ copies. In this work, we show that given a mixed state $$\rho \in \mathbb {C}^{d \times d}$$ : Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov’s algebra of polynomial functions on Young diagrams.

We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set $$J_{P}$$ these imply that periodic cutpoints of some invariant subcontinua of $$J_{P}$$ are also cutpoints of $$J_{P}$$ . We deduce that, under certain assumptions on invariant subcontinua Q of $$J_{P}$$ , every Riemann ray to Q landing at a periodic repelling/parabolic point $$x\in Q$$ is isotopic to a Riemann ray to $$J_{P}$$ relative to Q.

A three-dimensional Direct Numerical Simulation (DNS) database of statistically planar $$H_{2} -$$ H 2 - air turbulent premixed flames with an equivalence ratio of 0.7 spanning a large range of Karlovitz number has been utilised to assess the performances of the extrapolation relations, which approximate the stretch rate and curvature dependences of density-weighted displacement speed $$S_{d}^{*}$$ S d ∗ . It has been found that the correlation between $$S_{d}^{*}$$ S d ∗ and curvature remains negative and a significantly non-linear interrelation between $$S_{d}^{*}$$ S d ∗ and stretch rate has been observed for all cases considered here. Thus, an extrapolation relation, which assumes a linear stretch rate dependence of density-weighted displacement speed has been found to be inadequate. However, an alternative extrapolation relation, which assumes a linear curvature dependence of $$S_{d}^{*}$$ S d ∗ but allows for a non-linear stretch rate dependence of $$S_{d}^{*}$$ S d ∗ , has been found to be more successful in capturing local behaviour of the density-weighted displacement speed. The extrapolation relations, which express $$S_{d}^{*}$$ S d ∗ as non-linear functions of either curvature or stretch rate, have been found to capture qualitatively the non-linear curvature and stretch rate dependences of $$S_{d}^{*}$$ S d ∗ more satisfactorily than the linear extrapolation relations. However, the improvement comes at the cost of additional tuning parameter. The Markstein lengths LM for all the extrapolation relations show dependence on the choice of reaction progress variable definition and for some extrapolation relations LM also varies with the value of reaction progress variable. The predictions of an extrapolation relation which involve solving a non-linear equation in terms of stretch rate have been found to be sensitive to the initial guess value, whereas a high order polynomial-based extrapolation relation may lead to overshoots and undershoots. Thus, a recently proposed extrapolation relation based on the analysis of simple chemistry DNS data, which explicitly accounts for the non-linear curvature dependence of the combined reaction and normal diffusion components of $$S_{d}^{*}$$ S d ∗ , has been shown to exhibit promising predictions of $$S_{d}^{*}$$ S d ∗ for all cases considered here.

Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $\alpha\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $\mu$ has polynomial growth, we give a sufficient condition for $L^2(\mu)$-boundedness of singular integral operators with smooth odd kernels of convolution type., Comment: 40 pages; removed the measurability assumption from Theorem 1.4, many minor improvements; to appear in Rev. Mat. Iberoam

One of the most significant problems in oil and gas sector is the swelling of shale when it comes in contact with water. The migration of hydrogen ions (H+) from the water-based drilling fluid into the platelets of shale formation causes it to swell, which eventually increases the size of the shale sample and makes it structure weak. This contact results in the wellbore instability problem that ultimately reduces the integrity of a wellbore. In this study, the swelling of a shale formation was modeled using the potential of first order kinetic equation. Later, to minimize its shortcoming, a new proposed model was formulated. The new model is based on developing a third degree polynomial equation that is used to model the swelling percentages obtained through linear dynamic swell meter experiment performed on a shale formation when it comes in contact with a drilling fluid. These percentages indicate the hourly change in sample size during the contact. The variables of polynomial equation are dependent on the time of contact between the mud and the shale sample, temperature of the environment, clay content in shale and experimental swelling percentages. Furthermore, the equation also comprises of adjustable parameters that are fine-tuned in such a way that the polynomial function is best fitted to the experimental datasets. The MAE (mean absolute error) of the present model, namely Scaling swelling equation was found to be 2.75%, and the results indicate that the Scaling Swelling equation has the better performance than the first order kinetics in terms of swelling predication. Moreover, the proposed model equation is also helpful in predicting the swelling onset time when the mud and shale comes in direct contact with each other. In both the cases, the percentage deviation in predicting the swelling initiation time is close to 10%. This information will be extremely helpful in forecasting the swelling tendency of shale sample in a particular mud. Also, it helps in validating the experimental results obtained from linear dynamic swell meter.

Let $\mathrm{R}$ be a real closed field. We prove that for each fixed $\ell, d \geq 0$, there exists an algorithm that takes as input a quantifier-free first order formula $��$ with atoms $P=0, P > 0, P < 0 \text{ with } P \in \mathcal{P} \subset \mathrm{D}[X_1,\ldots,X_k]^{\mathfrak{S}_k}_{\leq d}$, where $\mathrm{D}$ is an ordered domain contained in $\mathrm{R}$, and computes the ranks of the first $(\ell+1)$ cohomology groups, of the symmetric semi-algebraic set defined by $��$. The complexity of this algorithm (measured by the number of arithmetic operations in $\mathrm{D}$) is bounded by a \emph{polynomial} in $k$ and $\mathrm{card}(\mathcal{P})$ (for fixed $d$ and $\ell$). This result contrasts with the $\mathbf{PSPACE}$-hardness of the problem of computing just the zero-th Betti number (i.e. the number of semi-algebraically connected components) in the general case for $d \geq 2$ (taking the ordered domain $\mathrm{D}$ to be equal to $\mathbb{Z}$). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. We prove that the Specht modules corresponding to partitions having long lengths cannot occur with positive multiplicity in the isotypic decompositions of low dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks, and is the key technical tool in the design of the algorithm mentioned in the previous paragraph., 58 pages. Final version published in Found. Comput. Math

In secure distributed matrix multiplication (SDMM) the multiplication $\boldsymbol A \boldsymbol B$ from two private matrices $\boldsymbol A$ and $\boldsymbol B$ is outsourced to $N$ distributed servers. In $\ell $ -SDMM, the goal is to design a joint communication-computation procedure that optimally balances conflicting communication and computation metrics without leaking any information on both $\boldsymbol A$ and $\boldsymbol B$ to any set of $\ell { servers. To this end, the user applies coding with $\tilde { \boldsymbol A}_{i}$ and $\tilde { \boldsymbol B}_{i}$ representing encoded versions of $\boldsymbol A$ and $\boldsymbol B$ destined to the $i$ -th server. Now, SDMM involves multiple tradeoffs. One such tradeoff is the tradeoff between upload (UL) and download (DL) costs. To find a good balance between these two metrics, we propose two schemes which we term USCSA and GSCSA that are based on secure cross subspace alignment (SCSA). We show that there are various scenarios where they outperform existing SDMM schemes from the literature with respect to UL-DL efficiency. Next, we implement schemes from the literature, including USCSA and GSCSA, and test their performance on Amazon EC2. Our numerical results show that USCSA and GSCSA establish a good balance between the time spent on the communication and computation in SDMMs. This is because they combine advantages of polynomial codes, namely low time for the upload of $\tilde { \boldsymbol A}_{i}$ and $\tilde { \boldsymbol B}_{i}$ , $i\in [1:N]$ , and the computation of $\boldsymbol O_{i}=\tilde { \boldsymbol A}_{i}\tilde { \boldsymbol B}_{i}$ , with those of SCSA, being a low timing overhead for the download of $\boldsymbol O_{i}$ , $i\in [1:N]$ , and the decoding of $\boldsymbol A \boldsymbol B$ .

In this paper, the representations of meromorphic solutions for three types of non-linear difference equations of form \begin{document}$ f^{n}(z)+P_{d}(z, f) = u(z)e^{v(z)}, $\end{document} \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\lambda z}+p_{2}e^{-\lambda z} $\end{document} and \begin{document}$ f^{n}(z)+P_{d}(z, f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z} $\end{document} are investigated, where $ n\geq 2 $ is an integer, $ P_{d}(z, f) $ is a difference polynomial in $ f $ of degree $ d\leq n-1 $ with small coefficients, $ u(z) $ is a non-zero polynomial, $ v(z) $ is a non-constant polynomial, $ \lambda, p_{j}, \alpha_{j}\; (j = 1, 2) $ are non-zero constants. Some examples are also presented to show our results are best in certain sense.

The production of polyacrylonitrile (PAN)-based carbon fiber includes two major steps, stabilization (oxidization) and carbonization. Stabilization is a slow and energy-consuming process and involves using a series of ovens that need to be carefully tuned to achieve desirable quality at the lowest cost. This usually involves meeting fiber density or cyclization levels at each zone of the oxidation furnace, often achieved through trial and error by a team of skilled operators. In this work, a simple empirical model was developed to predict density from oven system parameters and cyclization onset temperature of the PAN feedstock, parametrized as Sn, representing the oxidation depth. The relative cyclization index (RCI) and density of the fibers were analyzed after various oxidation treatments and were used to derive the model parameters. The RCI of stabilized fiber was well represented by a logistic function incorporating Sn while the fiber density was well predicted by a polynomial function of RCI. The model was verified using stabilization trials conducted on the same precursor in a continuous oven system and in a batch oven, and then on a second commercial precursor using the batch oven. Statistical analysis on these trials shows good agreement with model predictions.

The b-Exact Multicover problem takes a universe U of n elements, a family $\mathcal {F}$ of m subsets of U, a function ${\textsf {dem}}: U \rightarrow \{1,\ldots ,b\}$ and a positive integer k, and decides whether there exists a subfamily(set cover) $\mathcal {F}^{\prime }$ of size at most k such that each element u ∈ U is covered by exactly dem(u) sets of $\mathcal {F}^{\prime }$ . The b-Exact Coverage problem also takes the same input and decides whether there is a subfamily $\mathcal {F}^{\prime } \subseteq \mathcal {F}$ such that there are at least k elements that satisfy the following property: u ∈ U is covered by exactly dem(u) sets of $\mathcal {F}^{\prime }$ . Both these problems are known to be NP-complete. In the parameterized setting, when parameterized by k, b-Exact Multicover is W[1]-hard even when b = 1. While b-Exact Coverage is FPT under the same parameter, it is known to not admit a polynomial kernel under standard complexity-theoretic assumptions, even when b = 1. In this paper, we investigate these two problems under the assumption that every set satisfies a given geometric property π. Specifically, we consider the universe to be a set of n points in a real space $\mathbb {R}^{d}$ , d being a positive integer. When d = 2 we consider the problem when π requires all sets to be unit squares or lines. When d > 2, we consider the problem where π requires all sets to be hyperplanes in $\mathbb {R}^{d}$ . These special versions of the problems are also known to be NP-complete. When parameterized by k, the b-Exact Coverage problem has a polynomial size kernel for all the above geometric versions. The b-Exact Multicover problem turns out to be W[1]-hard for squares even when b = 1, but FPT for lines and hyperplanes. Further, we also consider the b-Exact Max. Multicover problem, which takes the same input and decides whether there is a set cover $\mathcal {F}^{\prime }$ such that every element u ∈ U is covered by at least dem(u) sets and at least k elements satisfy the following property: u ∈ U is covered by exactly dem(u) sets of $\mathcal {F}^{\prime }$ . To the best of our knowledge, this problem has not been studied before, and we show that it is NP-complete (even for the case of lines). In fact, the problem turns out to be W[1]-hard in the general setting, when parameterized by k. However, when we restrict the sets to lines and hyperplanes, we obtain FPT algorithms.

UDC 517.5 For the rational function $r(z)=p(z)/H(z)$ having all its zeros in $|z|\leq 1,$ it is known that\begin{equation*}|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{for}\quad |z|=1,\end{equation*}where $H(z)=\prod_{j=1}^n(z - c_j),$ $|c_j|>1,$ $n$ is a positive integer, $B(z)=H^*(z)/H(z),$ and $H^*(z)=z^n\overline{H(1/\overline{z})}.$In this paper, we improve the above mentioned inequality for the rational function $r(z)$ with all zeros in $|z|\leq 1$ and a zero of order $s$ at the origin. Our main results refine and generalize some known rational inequalities.

We consider the spaces $${\mathcal {F}}_\mu$$ of polynomial $$\mu$$ -densities on the line as $${\mathfrak {sl}}(2)$$ -modules and then we compute the cohomological spaces $$\text {H}^1_\text {diff}({\mathfrak {sl}}(2), {\mathcal {D}}_{{\bar{\lambda }},\mu })$$ , where $$\mu \in {\mathbb {R}}$$ , $${\bar{\lambda }}=(\lambda _1,\dots ,\lambda _n) \in {\mathbb {R}}^n$$ and $${\mathcal {D}}_{{\bar{\lambda }},\mu }$$ is the space of n-ary differential operators from $${\mathcal {F}}_{\lambda _1}\otimes \cdots \otimes {\mathcal {F}}_{\lambda _n}$$ to $${\mathcal {F}}_\mu$$ .

The classical Weyl Law says that if NM(λ) denotes the number of eigenvalues of the Laplace operator on a d-dimensional compact manifold M without a boundary that are less than or equal to λ, then $${N_M}(\lambda ) = c{\lambda ^d} + O({\lambda ^{d - 1}}).$$ This paper explores the prospects of improvements of Weyl remainders on products of manifolds. In particular we obtain a polynomial improvement to the Weyl remainder for products of spheres, demonstrate how Duistermaat and Giullemin’s result implies a little-o improvement to the remainder for products of compact Riemannian manifolds without boundary, and conjecture that polynomial improvements hold for these more general products.

Ultrasound image of B-mode is reconstructed by DC(Direct Current) removal, quadrature demodulation, low-pass filtering, envelope detection and scanning conversion. Demodulation is the most important of the above. Because of the diversity of human organs and their corresponding acoustic properties, the shift frequency of echo signal changes as the depth increases. For a more accurate recovery signal, segmented function is used to describe the process. Traditionally, polynomial fitting or smooth spline was used to eliminate the discontinuity between the neighbor segment. In this brief, a phase-based quick-fitting method is proposed to optimize the step. Experimental investigation revealed that the phase fitting is applied to the linear array, and the mean-variance is 0.0002( $MHz^{2}$ ) compared with the smooth spline( $0.0033MHz^{2}$ ). For convex probe, the mean-variance( $0.0005MHz^{2}$ ) is also smaller than that of polynomial fitting( $0.0046MHz^{2}$ ). Otherwise, this algorithm can overcome the boundary oscillation caused by frequency fitting. Especially, the results are stable even if the number of segments is large. Complex calculations are not required, the process can be implemented regardless of software and hardware.

Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.

We consider a multiple-depots extension of the classic Hamiltonian path problem where k salesmen are initially located at different depots. To the best of our knowledge, no algorithm for this problem with a constant approximation ratio has been previously proposed, except for some special cases. We present a polynomial algorithm with a tight approximation ratio of $$\max \left\{ \frac{3}{2},2-\frac{1}{k}\right\} $$ for arbitrary $$k\geqslant 1$$ , and an algorithm with approximation ratio $$\frac{5}{3}$$ that runs in polynomial time for fixed k. Moreover, we develop a recursive framework to improve the approximation ratio to $$\frac{3}{2}+\epsilon $$ . This framework is polynomial for fixed k and $$\epsilon $$ , and may be useful in improving the Christofides-like heuristics for other related multiple salesmen routing problems.

Let G = GL(m|n) be a general linear supergroup, and Gev be its even subsupergroup isomorphic to GL(m) ×GL(n). In this paper, we use the explicit description of Gev-primitive vectors in the costandard supermodule ∇(λ), the largest polynomial G-subsupermodule of the induced supermodule ${H^{0}_{G}}(\lambda )$ , for (m|n)-hook partition λ, and properties of certain morphisms ψk to derive results related to odd linkage for G over a field F of characteristic different from 2.

Let p(z) be a polynomial of degree n and $$\alpha$$ be any real or complex number, the polar derivative of p(z), denoted by $$D_{\alpha }p(z)$$ , is defined as $$\begin{aligned} D_{\alpha }p(z)=np(z)+(\alpha -z)p^{'}(z). \end{aligned}$$ Let $$p(z)=a_{0}+\sum ^{n}_{\nu =\mu }a_{\nu }z^{\nu }$$ , $$1\le \mu \le n$$ , be a polynomial of degree n having no zeros in $$|z|0$$ , then for every real or complex number $$\alpha$$ with $$|\alpha |\ge R$$ and $$00$$ . Further, we extend another improvement of the above inequality proved by them into $$L^{p}$$ -norm.