Monomial Functions Research Articles

Adaptive asymptotic solutions of inflationary models in the Hamilton-Jacobi formalism: application to T-models

We develop a method to compute the slow-roll expansion for the Hubble parameter in inflationary models in a flat Friedmann-Lemaître-Robertson-Walker spacetime that is applicable to a wide class of potentials including monomial, polynomial, or rational functions of the inflaton, as well as polynomial or rational functions of the exponential of the inflaton. The method, formulated within the Hamilton-Jacobi formalism, adapts the form of the slow-roll expansion to the analytic form of the inflationary potential, thus allowing a consistent order-by-order computation amenable to Padé summation. Using T-models as an example, we show that Padé summation extends the domain of validity of this adapted slow-roll expansion to the end of inflation. Likewise, Padé summation extends the domain of validity of kinetic-dominance asymptotic expansions of the Hubble parameter into the fast-roll regime, where they can be matched to the aforesaid Padé-summed slow-roll expansions. This matching in turn determines the relation between the expansions for the number N of e-folds and allows us to compute the total amount of inflation as a function of the initial data or, conversely, to select initial data that correspond to a fixed total amount of inflation. Using the slow-roll stage expansions, we also derive expansions for the corresponding spectral index ns accurate to order 1/N2, and tensor-to-scalar ratio r accurate to order 1/N3 for these T-models.

Bent functions satisfying the dual bent condition and permutations with the $$(\mathcal {A}_m)$$ property

AbstractThe concatenation of four Boolean bent functions $$f=f_1||f_2||f_3||f_4$$ f = f 1 | | f 2 | | f 3 | | f 4 is bent if and only if the dual bent condition $$f_1^* + f_2^* + f_3^* + f_4^* =1$$ f 1 ∗ + f 2 ∗ + f 3 ∗ + f 4 ∗ = 1 is satisfied. However, to specify four bent functions satisfying this duality condition is in general quite a difficult task. Commonly, to simplify this problem, certain relations between $$f_i$$ f i are assumed, as well as functions $$f_i$$ f i of a special shape are considered, e.g., $$f_i(x,y)=x\cdot \pi _i(y)+h_i(y)$$ f i ( x , y ) = x · π i ( y ) + h i ( y ) are Maiorana-McFarland bent functions. In the case when permutations $$\pi _i$$ π i of $$\mathbb {F}_2^m$$ F 2 m have the $$(\mathcal {A}_m)$$ ( A m ) property and Maiorana-McFarland bent functions $$f_i$$ f i satisfy the additional condition $$f_1+f_2+f_3+f_4=0$$ f 1 + f 2 + f 3 + f 4 = 0 , the dual bent condition is known to have a relatively simple shape allowing to specify the functions $$f_i$$ f i explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions $$f_i$$ f i satisfy the condition $$f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)$$ f 1 ( x , y ) + f 2 ( x , y ) + f 3 ( x , y ) + f 4 ( x , y ) = s ( y ) and provide a construction of new permutations with the $$(\mathcal {A}_m)$$ ( A m ) property from the old ones. Combining these two results, we obtain a recursive construction method of bent functions satisfying the dual bent condition. Moreover, we provide a generic condition on the Maiorana-McFarland bent functions $$f_1,f_2,f_3,f_4$$ f 1 , f 2 , f 3 , f 4 stemming from the permutations of $$\mathbb {F}_2^m$$ F 2 m with the $$(\mathcal {A}_m)$$ ( A m ) property, such that the concatenation $$f=f_1||f_2||f_3||f_4$$ f = f 1 | | f 2 | | f 3 | | f 4 does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations $$\pi _i$$ π i of $$\mathbb {F}_{2^m}$$ F 2 m with the $$(\mathcal {A}_m)$$ ( A m ) property and monomial functions $$h_i$$ h i on $$\mathbb {F}_{2^m}$$ F 2 m , we provide explicit constructions of such bent functions; a particular case of our result shows how one can construct bent functions from APN permutations, when m is odd. Finally, with our construction method, we explain how one can construct homogeneous cubic bent functions, noticing that only very few design methods of these objects are known.

James reduced product schemes and double quasisymmetric functions

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product JCP∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for JCP∞ in both cohomology and K-theory.Here, we study the torus-equivariant cohomology of JCP∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis.Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety.

Error analysis of the element-free Galerkin method for a nonlinear plate problem

In this work, we derive a geometrically nonlinear plate model based on the Kirchhoff hypothesis and the large deflection hypothesis, and provide an error analysis of the corresponding element-free Galerkin method. The penalty method is employed to enforce boundary conditions. The error estimate explicitly depends on nodal spacing, number of monomial basis functions, continuity of weight functions, and penalty factors, which provides some practical choices among these key parameters in engineering applications. In addition, we offer guidance on selecting appropriate penalty factors to improve numerical accuracy. Numerical experiments, involving clamped square and circle plates with uniform and nonuniform nodes, as well as a plate model with a corner singularity, are presented to validate the theoretical results.

Monomial Boolean functions with large high-order nonlinearities

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher order nonlinearities of some monomial Boolean functions.We prove lower bounds on the second-order nonlinearities of functions trn(x7) and trn(x2r+3) where n=2r. Among all monomial Boolean functions, our bounds match the best second-order nonlinearity lower bounds by Carlet [IEEE Transactions on Information Theory 54(3), 2008] and Yan and Tang [Discrete Mathematics 343(5), 2020] for odd and even n, respectively. We prove a lower bound on the third-order nonlinearity for functions trn(x15), which is the best third-order nonlinearity lower bound. For any r, we prove that the r-th order nonlinearity of trn(x2r+1−1) is at least 2n−1−2(1−2−r)n+r2r−1−1−O(2n2). For r≪log2⁡n, this is the best lower bound among all explicit functions.

Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-Commuting Variables

We introduce a new powersum basis for the Hopf algebra of quasisymmetric functions that refines the powersum symmetric basis. Unlike the quasisymmetric powersums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum P basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.

Numerical solution of Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems using fractional-order Legendre collocation method

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton’s portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton’s portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation.

Systematic Design of a Pseudodifferential VCO Using Monomial Fitting

Digital integrated electronics benefits from its higher abstraction level, allowing optimisation methods and automated workflows. However, analogue integrated circuit design is still predominantly done manually, leading to lengthy design cycles. This paper proposes a new systematic design approach for the sizing of analogue integrated circuits to address this issue. The method utilises a surrogate optimisation technique that approximates a simple monomial function based on few simulation results. These monomials are convex and can be optimised using a simple linear optimisation routine, resulting in a single global optimal solution. We show that monomial functions, in many cases, have an analytic relation to integrated circuits, making them well suited for the application. The method is demonstrated by designing a 14 MHz pseudodifferential voltage-controlled oscillator (VCO) with minimised current consumption and is manufactured in a 180 nm process. The measured total current matches the predicted and is lower than that for other similar state-of-the-art VCOs.

Short [formula omitted]-rotation symmetric Boolean functions

Rotation symmetric Boolean functions were introduced by Pieprzyk and Qu in the late 1990’s. They showed that these functions are useful, among other things, in the design of fast hashing algorithms with strong cryptographic properties. The concept of rotation symmetric Boolean functions has been generalized to a class of functions known as k-rotation Boolean functions, where k divides n and n is the number of variables of the Boolean function. Analogous to the case of regular rotation symmetric Boolean functions, a monomial k-rotation Boolean function is called long k-cycle if the number of terms coincides with n/k and short k-cycle if the number of terms is less than n/k. In this work we characterize short k-cycles by providing specific generators for them. We also provide a count for the number of short k-cycles in n variables for a fixed degree and for a specified length.

A new class of generalized almost perfect nonlinear monomial functions

In this short note, we present a new class of GAPN power functions of the type xk2p2i+k1pi+k0 over finite fields Fpn with p odd and gcd(n,i)=1 (up to EA-equivalence).

Monomial functions, normal polynomials and polynomial equations

In this paper we consider generalized monomial functions f, g:{mathbb {F}}rightarrow {mathbb {C}} (of possibly different degree) that also fulfill f(P(x))=Q(g(x))x∈F,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f(P(x))= Q(g(x)) \\qquad \\left( x\\in {\\mathbb {F}}\\right) , \\end{aligned}$$\\end{document}where Pin {mathbb {F}}[x] and Qin {mathbb {C}}[x] are given (classical) polynomials.

Normal bundle of monomial curves: an application to rational curves

In this note, we give an application to the study of general rational curves in \mathbb{P}^s(\mathbb{C}) of the calculation of the splitting type of the normal bundle of any smooth monomial rational curve (i.e., embedded by monomial functions).

Minimization Fractional Prophet Inequalities for Sequential Procurement

We consider a minimization variant on the classical prophet inequality with monomial cost functions. A firm would like to procure some fixed amount of a divisible commodity from sellers that arrive sequentially. Whenever a seller arrives, the seller’s cost function is revealed, and the firm chooses how much of the commodity to buy. We first show that if one restricts the set of distributions for the coefficients to a family of natural distributions that include, for example, the uniform and truncated normal distributions, then there is a thresholding policy that is asymptotically optimal in the number of sellers. We then compare two scenarios based on whether the firm has in-house production capabilities or not. We precisely compute the optimal algorithm’s competitive ratio when in-house production capabilities exist and for a special case when they do not. We show that the main advantage of the ability to produce the commodity in house is that it shields the firm from price spikes in worst-case scenarios. Funding: This work was supported by NSF Grants [CNS-2146814, CPS-2136197, CNS-2106403, NGSDI-2105648].

Minimization Fractional Prophet Inequalities for Sequential Procurement

We consider a minimization variant on the classical prophet inequality with monomial cost functions. A firm would like to procure some fixed amount of a divisible commodity from sellers that arrive sequentially. Whenever a seller arrives, the seller’s cost function is revealed, and the firm chooses how much of the commodity to buy. We first show that if one restricts the set of distributions for the coefficients to a family of natural distributions that include, for example, the uniform and truncated normal distributions, then there is a thresholding policy that is asymptotically optimal in the number of sellers. We then compare two scenarios based on whether the firm has in-house production capabilities or not. We precisely compute the optimal algorithm’s competitive ratio when in-house production capabilities exist and for a special case when they do not. We show that the main advantage of the ability to produce the commodity in house is that it shields the firm from price spikes in worst-case scenarios. Funding: This work was supported by NSF Grants [CNS-2146814, CPS-2136197, CNS-2106403, NGSDI-2105648].

A general construction of permutation polynomials of [formula omitted

Let r be a positive integer, h(X)∈Fq2[X], and μq+1 be the subgroup of order q+1 of Fq2⁎. It is well known that Xrh(Xq−1) permutes Fq2 if and only if gcd(r,q−1)=1 and Xrh(X)q−1 permutes μq+1. There are many ad hoc constructions of permutation polynomials of Fq2 of this type such that h(X)q−1 induces monomial functions on the cosets of a subgroup of μq+1. We give a general construction that can generate, through an algorithm, all permutation polynomials of Fq2 with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.

Improving the Efficiency of Hedge Trading Using Higher-Order Standardized Weather Derivatives for Wind Power

Since the future output of wind power generation is uncertain due to weather conditions, there is an increasing need to manage the risks associated with wind power businesses, which have been increasingly implemented in recent years. This study introduces multiple weather derivatives of wind speed and temperature and examines their effectiveness in reducing (hedging) the fluctuation risk of future cash flows attributed to wind power generation. Given the diversification of hedgers and hedging needs, we propose new standardized derivatives with higher-order monomial payoff functions, such as “wind speed cubic derivatives” and “wind speed and temperature cross-derivatives,” to minimize the cash flow variance and develop a market-trading scheme to practically use these derivatives in wind power businesses. In particular, while demonstrating the importance of standardizing weather derivatives regarding market liquidity and efficiency, we propose a strategy to narrow down the required number (or volume) of traded instruments and improve trading efficiency by utilizing the least absolute shrinkage and selection operator (LASSO) regression. Empirical analysis reveals that higher-order, multivariate standardized derivatives can not only enhance the out-of-sample hedge effect but also help reduce trading volume. The results suggest that diversification of hedging instruments increases transaction flexibility and helps wind power generators find more efficient portfolios, which can be generalized to risk management practices in other businesses.

Beurling’s theorem for the Hardy operator on $$L^2[0,1]$$

We prove that the invariant subspaces of the Hardy operator on $$L^2[0,1]$$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.

Superimposed hydration exothermic model of cement slurry considering different reaction rates of various active substances

Cement components usually contain two or more hydration active substances with different hydration reaction rates, and the existing hydration exothermic model of cement-based materials do not consider the difference of reaction rates of active substances. This paper proposes a new exothermic hydration model of cement-based materials, which takes into consideration the different reaction rates of multi-component active substances. Firstly, the hydration heat and the representative temperatures of cement slurry with the water-cement ratio of 0.3, are tested through adiabatic and dissolution heat methods. Secondly, the total hydration reaction process is assumed as the composition of the reaction processes of multi-component active substances. Then, a hydration exothermic model witch is superimposed by several exponential functions is fit according to the exothermic hydration experimental data and the Arrhenius chemical expression. The comparison results show that the proposed model is more accurate than the monomial exponential function model, especiallyat the early age before the initial setting time. In conclusion, the proposed model can better explain the influence of the reaction rate difference of different active substances of cement on the exothermic process, and the accuracy of hydration analysis is more accurate by using it than others.

Several classes of exponential sums and three-valued Walsh spectrums over finite fields

Let r≡1(mod4) be a prime, m a positive integer, ϕ(rm)2 the multiplicative order of 2 modulo rm, and q=2ϕ(rm)2, where ϕ(⋅) is the Euler's function. Let γ be a primitive element of the finite field Fq and α=γq−1rm. In this paper, we shall explicitly calculate the exponential sums S(a)=∑i=0rm−1χ(aαi), a∈Fq, where χ is the canonical additive character of Fq. As applications, using elliptic curves and cyclotomic numbers over Fr, the three-valued Walsh spectrums of four classes of monomial functions fa(x)=Trq/2(axq−1rm) are determined.

Asymptotic Müntz–Szász theorems

We define a monomial space to be a subspace of L$^2([0,1])$ that can be approximated by spaces that are spanned by monomial functions. We describe the structure of monomial spaces.