Estimating temperature fluctuations in the early universe

A lagrangian for a k-essence field is constructed for a constant scalar potential, and its form is determined when the scale factor is very small as compared to the present epoch but very large as compared to the inflationary epoch. This means that one is already in an expanding and flat universe. The form is similar to that of an oscillator with time-dependent frequency. Expansion is naturally built into the theory with the existence of growing classical solutions of the scale factor. The formalism allows one to estimate the temperature fluctuations of the background radiation at these early stages (as compared to the present epoch) of the Universe. If the temperature is T _a at time t _a and T _b at time t _b (t _b > t _a ), then, for small times, the probability evolution for the logarithm of the inverse temperature can be estimated as

A transformation of series

The transformation (*) $$\sum\limits_{\nu = 0}^\infty {t_\nu z^\nu \to } \sum\limits_{\nu = 0}^\infty {\left\{ {\sum\limits_{\tau = 0}^{h - 1} {z^\tau } \Delta ^\nu t_{h\nu + \tau } + \frac{{z^h }}{{1 - z}}\Delta ^\nu t_{h(\nu + 1)} } \right\}} \left( {\frac{{z^{h + 1} }}{{1 - z}}} \right)^\nu$$ whereh≥0 is an integer and Δ operates upon the coefficients {t _v } of the series being transformed, is derived. Whenh=0, the above transformation is the generalised Euler transformation, of which (*) is itself a generalisation. Based upon the assumption that \(t_\nu = \int\limits_0^1 {\varrho ^\nu d\sigma (\varrho ) } (\nu = 0, 1,...)\) , where σ(?) is bounded and non-decreasing for 0≤?≤1 and subject to further restrictions, a convergence theory of (*) is given. Furthermore, the question as to when (*) functions as a convergence acceleration transformation is investigated. Also the optimal valne ofh to be taken is derived. A simple algorithm for constructing the partial sums of (*) is devised. Numerical illustrations relating to the case in whicht _v =(v+1) ^?1 (v=0,1,...) are given.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

Representation of ℤ<Subscript>4</Subscript>-Linear Preparata Codes Using Vector Fields

A binary code is called ℤ₄-linear if its quaternary Gray map preimage is linear. We show that the set of all quaternary linear Preparata codes of length n = 2^m, m odd, m ≥ 3, is nothing more than the set of codes of the form with

Large deviations for distributions of sums of random variables: Markov chain method

Let {ξ _k } _k=0^∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} , n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| ^p , p > 0, and exponential random variables with g(x) = x for x ≥ 0.     

Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor

The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:

The exact region of stability for MacCormack scheme

Let the two dimensional scalar advection equation be given by $$\frac{{\partial u}}{{\partial t}} = \hat a\frac{{\partial u}}{{\partial x}} + \hat b\frac{{\partial u}}{{\partial y}}.$$ We prove that the stability region of the MacCormack scheme for this equation isexactly given by $$\left( {\hat a\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} + \left( {\hat b\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} \leqslant 1$$ where Δ _t , Δ _x and Δ _y are the grid distances. It is interesting to note that the stability region is identical to the one for Lax-Wendroff scheme proved by Turkel.     

Small deviations for two classes of Gaussian stationary processes and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-functionals, 0 < <Emphasis Type="Italic">p</Emphasis> ≤ ∞

Let w(t) be a standard Wiener process, w(0) = 0, and let η _a (t) = w(t + a) − w(t), t ≥ 0, be increments of the Wiener process, a > 0. Let Z _a (t), t ∈ [0, 2a], be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form E Z _a (t)Z _a (s) = 1/2[a − |t − s|], t, s ∈ [0, 2a]. For 0 < p < ∞, we prove results on sharp asymptotics as ɛ → 0 of the probabilities

A Sigma-Pi-Sigma Neural Network (SPSNN)

This letter presents a sigma-pi-sigma neural network (SPSNN) structure. The SPSNN can learn to implement static mapping that multilayer neural networks and radial basis function networks usually do. The output of the SPSNN has the sum of product-of-sum form , where x _j's are inputs, N _v is the number of inputs, f _nij() is a function to be generated through the network training, and K is the number of pi-sigma network (PSN) which is the basic building block for SPSNN. A linear memory array can be used to implement f _nij (). The function f _nij (x _j ) can be expressed as , where B _ijk() is a single-variable basis function, w _nijk's are weight values stored in memory, N _q is the quantized element number for x _j , and N _e is the number of basis functions in the neighborhood used for storing information for x _j. If all B _ijk()'s are Gaussian functions, the new neural network degenerates to a Gaussian function network. This paper focuses on the use of overlapped rectangular pulses as the basis functions. With such basis functions, will equal either zero or w _nijk, and the computation of f _nij (x _j) becomes a simple addition of retrieved w _nijk's. The new neural network structure demonstrates excellent learning convergence characteristics and requires small memory space. It has merits over multilayer neural networks, radial basis function networks and CMAC. This revised version was published online in June 2006 with corrections to the Cover Date.     

Computation of Battle–Lemarie Wavelets Using an FFT-Based Algorithm

Battle and Lemarie derived independently wavelets by orthonormalizing B-splines. The scaling function _m (t) corresponding to Battle–Lemarie's wavelet _m (t) is given by , where B _m(t) is the mth-order central B-spline and the coefficients _{m, k} satisfy . In this paper, we propose an FFT-based algorithm for computing the expansion coefficients _{m, k} and the two-scale relations of the scaling functions and wavelets. The algorithm is very simple and it can be easily implemented. Moreover, the expansion coefficients can be efficiently and accurately obtained via multiple sets of FFT computations. The computational approach presented in this paper here is noniterative and is more efficient than the matrix approach recently proposed in the literature.     

Computational Divided Differencing and Divided-Difference Arithmetics

Tools for computational differentiation transform a program that computes a numerical function F(x) into a related program that computes F(x) (the derivative of F). This paper describes how techniques similar to those used in computational-differentiation tools can be used to implement other program transformations—in particular, a variety of transformations for computational divided differencing. The specific technical contributions of the paper are as follows:– It presents a program transformation that, given a numerical function F(x) defined by a program, creates a program that computes F[x ₀, x ₁], the first divided difference of F(x), where – It shows how computational first divided differencing generalizes computational differentiation.– It presents a second program transformation that permits the creation of higher-order divided differences of a numerical function defined by a program.– It shows how to extend these techniques to handle functions of several variables.The paper also discusses how computational divided-differencing techniques could lead to faster and/or more robust programs in scientific and graphics applications.Finally, the paper describes how computational divided differencing relates to the numerical-finite-differencing techniques that motivated Robert Paige's work on finite differencing of set-valued expressions in SETL programs.     

On a direct algorithm for the generation of log-normal pseudo-random numbers

The random variable \(\left( {\prod {_{i = 1}^n {{X_i } \mathord{\left/ {\vphantom {{X_i } {X_{i + n} }}} \right. \kern-0em} {X_{i + n} }}} } \right)^{{1 \mathord{\left/ {\vphantom {1 {\sqrt {2n} }}} \right. \kern-0em} {\sqrt {2n} }}}\) is used to generate standard log-normal variables Λ(0, 1), where theX _i are independent uniform variables on [0, 1].     

Onn-column 0, 1-matrices with allk-projections surjective

This paper presents a new approach to construct a smalln-column 0, 1-matrix for two given integersn andk(k, such that everyk-column projection contains all 2 ^k possible row vectors, namely surjective on {0, 1} ^k . The number of the matrix's rows does not exceed . This approach has considerable advantage for smallk and practical sizes ofn. It can be applied to the test generation of VLSI circuits, the design of fault tolerant systems and other fields.     

On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups

In this paper, we discuss the minimal number of observables Q ₁, ..., Q , where expectation values at some time instants t ₁, ..., t _r determine the trajectory of a d-level quantum system (qudit) governed by the Gaussian semigroup . We assume that the macroscopic information about the system in question is given by the mean values E _j(Q _i) = tr(Q _i(t _j)) of n selfadjoint operators Q ₁, ..., Q _n at some time instants t ₁ < t ₂ < ... < t _r, where n < d ²– 1 and r deg (, ). Here (, ) stands for the minimal polynomial of the generator of the Gaussian flow (t).     

Behavior of confined fluids in nanoslit pores: the normal pressure tensor

The aim of our research is to develop a theory, which can predict the behavior of confined fluids in nanoslit pores. The nanoslit pores studied in this work consist of two structureless and parallel walls in the xy plane located at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a given uniform bulk density. We have derived the following general equation for prediction of the normal pressure tensor P _zz of confined inhomogeneous fluids in nanoslit pores:

A technical note on AC-Unification. The number of minimal unifiers of the equation αx1 + ⋯ + αxp ≐AC βy1 + ⋯ + βyq

We show in this note that the equation αx₁ + #x22EF; +αx_p?_ACβy₁ + α +βy_q where + is an AC operator and αx stands for x+...+x (α times), has exactly $$\left( { - 1} \right)^{p + q} \sum\limits_{i = 0}^p {\sum\limits_{j = 0}^q {\left( { - 1} \right)^{1 + 1} \left( {\begin{array}{*{20}c} p \\ i \\ \end{array} } \right)\left( {\begin{array}{*{20}c} q \\ j \\ \end{array} } \right)} 2^{\left( {\alpha + \begin{array}{*{20}c} {j - 1} \\ \alpha \\ \end{array} } \right)\left( {\beta + \begin{array}{*{20}c} {i - 1} \\ \beta \\ \end{array} } \right)} } $$ minimal unifiers if gcd(α, β)=1.     

Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ${\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, . In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A, B) and possibly on μ, T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation  A continuous function is of class , if it is strictly increasing and is of class if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function is said to be a class -function if, for any t ≥ 0, and for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L ₂-norm of matrices.     

On the complexity of computing determinants

We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in and bit operations; here denotes the largest entry in absolute value and the exponent adjustment by +o(1) captures additional factors for positive real constants C₁, C₂, C₃. The bit complexity results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n^3.2+o(1) and O(n^2.697263) ring additions, subtractions and multiplications.To B. David Saunders on the occasion of his 60th birthday     

On the variance of gaussian quadrature formulae in the ultraspherical case

For ultraspherical weight functions ω_λ(x)=(1–x²)^λ–1/2, we prove asymptotic bounds and inequalities for the variance Var(Q _n ^G ) of the respective Gaussian quadrature formulae Q _n ^G . A consequence for a large class of more general weight functions ω and the respective Gaussian formulae is the following asymptotic result, $$\mathop {lim}\limits_{n \to \infty } n \cdot Var\left( {Q_n^G } \right) = \pi \int_{ - 1}^1 {\omega ^2 \left( x \right)\sqrt {1 - x^2 } dx.} $$