Formule di quadratura di tipo gaussiano con valori delle derivate dell'integrando

The coefficientsA _hi (m,s) and the nodesx _i (m,s) for Gaussian-type quadrature formulae

$$\int\limits_{ - 1}^1 {f(x)dx = \mathop \sum \limits_{h = 0}^{2s} \mathop \sum \limits_{i = 1}^m } A_{hi} \cdot f^{(h)} (x_i )$$     

Numerical evaluation of a solution of a special mixed-type differential-difference equation

The function * $$f(t) = \frac{{e^{ - \alpha \gamma } }}{\pi }\int\limits_0^\infty {\cos t \xi e^{\alpha Ci(\xi )} \frac{{d\xi }}{{\xi ^\alpha }},t \in R,\alpha > 0} $$ [Ci(x)=cosine integral, γ=Euler's constant] is studied and numerically evaluated;f is a solution to the following mixed type differential-difference equation arising in applied probability: ** $$tf'(t) = (\alpha - 1)f(t) - \frac{\alpha }{2}[f(t - 1) + f(t + 1)]$$ satisfying the conditions: i) $$f(t) \geqslant 0,t \in R$$ , ii) $$f(t) = f( - t),t \in R$$ , iii) $$\int\limits_{ - \infty }^{ + \infty } {f(\xi )d\xi = 1} $$ . Besides the direct numerical evaluation of (*) and the derivation of the asymptotic behaviour off(t) fort→0 andt→∞, two different iterative procedures for the solution of (**) under the conditions (i) to (iii) are considered and their results are compared with the corresponding values in (*). Finally a Monte Carlo method to evaluatef(t) is considered.     

Fast hybrid matrix multiplication algorithms

New hybrid algorithms for matrix multiplication are proposed that have the lowest computational complexity in comparison with well-known matrix multiplication algorithms. Based on the proposed algorithms, efficient algorithms are developed for the basic operation \( D = C + \sum\limits_{l =1}^{\xi} A_{l} B_{l}\) of cellular methods of linear algebra, where A, B, and D are square matrices of cell size. The computational complexity of the proposed algorithms is estimated.     

An efficient technique for estimating the effect of small errors of control system implementation on the optimal program

The time-optimal problem is considered for a linear system with constant coefficients. For a piecewise constant program, a matrix equality of the form

$P_m^0 (T) = \sum\limits_{i = 0}^{m - 1} {\Delta C_{i,i + 1} (0)} \bullet e^{ - At_i } $

is obtained, where the left-hand side depends on the final point, while each term in the right-hand side depends only on one of the control switchings (t¡ are switching instants). This relation is fulfilled on the trajectories of the original system. The vector deviation from the final point for small errors of switching instants is found from this formula. Furthermore, simple procedures for calculating deviations in the case when the control and coefficient matrices are defined with errors are presented, a new method of solving the time-optimal problem is described, and the Pontryagin’s maximum principle is refined by adding the condition of getting into the final point. A numerical example is considered.

     

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

Two generalized Wigner–Yanase skew information and their uncertainty relations

In this paper, we first define two generalized Wigner–Yanase skew information \(|K_{\rho ,\alpha }|(A)\) and \(|L_{\rho ,\alpha }|(A)\) for any non-Hermitian Hilbert–Schmidt operator A and a density operator \(\rho \) on a Hilbert space H and discuss some properties of them, respectively. We also introduce two related quantities \(|S_{\rho ,\alpha }|(A)\) and \(|T_{\rho ,\alpha }|(A)\). Then, we establish two uncertainty relations in terms of \(|W_{\rho ,\alpha }|(A)\) and \(|\widetilde{W}_{\rho ,\alpha }|(A)\), which read

$$\begin{aligned}&|W_{\rho ,\alpha }|(A)|W_{\rho ,\alpha }|(B)\ge \frac{1}{4}\left| \mathrm {tr}\left( \left[ \frac{\rho ^{\alpha }+\rho ^{1-\alpha }}{2} \right] ^{2}[A,B]^{0}\right) \right| ^{2},\\&\sqrt{|\widetilde{W}_{\rho ,\alpha }|(A)| \widetilde{W}_{\rho ,\alpha }|(B)}\ge \frac{1}{4} \left| \mathrm {tr}\left( \rho ^{2\alpha }[A,B]^{0}\right) \mathrm {tr} \left( \rho ^{2(1-\alpha )}[A,B]^{0}\right) \right| . \end{aligned}$$

     

Cubature formulae for nearly singular and highly oscillating integrals

The paper deals with the approximation of integrals of the type

$$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$

where \({\mathrm {D}}=[-\,1,1]^2\), f is a function defined on \({\mathrm {D}}\) with possible algebraic singularities on \(\partial {\mathrm {D}}\), \({\mathbf {w}}\) is the product of two Jacobi weight functions, and the kernel \({\mathbf {K}}\) can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed.

     

High Order Algorithm for the Time-Tempered Fractional Feynman–Kac Equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in Wu et al. (Phys Rev E 93:032151, 2016), being called the time-tempered fractional Feynman–Kac equation named after Richard Feynman and Mark Kac who first considered the model describing the functional distribution of normal motion. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as

$$\begin{aligned} {^S\!}D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!=\!D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!-\!\lambda ^\gamma G(x,p,t) \end{aligned}$$

with \(\widetilde{\lambda }=\lambda + pU(x),\, p=\rho +J\eta ,\, J=\sqrt{-1}\), where

$$\begin{aligned} D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t) =\frac{1}{\varGamma (1-\gamma )} \left[ \frac{\partial }{\partial t}+\widetilde{\lambda } \right] \int _{0}^t{\left( t-z\right) ^{-\gamma }}e^{-\widetilde{\lambda }\cdot (t-z)}{G(x,p,z)}dz, \end{aligned}$$

and \(\lambda \ge 0\), \(0<\gamma <1\), \(\rho >0\), and \(\eta \) is a real number. The designed schemes are unconditionally stable and have the global truncation error \(\mathscr {O}(\tau ^2+h^2)\), being theoretically proved and numerically verified in complex space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the ‘physical’ equation (without artificial source term).

     

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.     

Clique Numbers of Random Subgraphs of Some Distance Graphs

We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:

$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$

where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.

     

An uncertainty relation in terms of generalized metric adjusted skew information and correlation measure

The uncertainty principle in quantum mechanics is a fundamental relation with different forms, including Heisenberg’s uncertainty relation and Schrödinger’s uncertainty relation. In this paper, we prove a Schrödinger-type uncertainty relation in terms of generalized metric adjusted skew information and correlation measure by using operator monotone functions, which reads,

$$\begin{aligned} U_\rho ^{(g,f)}(A)U_\rho ^{(g,f)}(B)\ge \frac{f(0)^2l}{k}\left| \mathrm {Corr}_\rho ^{s(g,f)}(A,B)\right| ^2 \end{aligned}$$

for some operator monotone functions f and g, all n-dimensional observables A, B and a non-singular density matrix \(\rho \). As applications, we derive some new uncertainty relations for Wigner–Yanase skew information and Wigner–Yanase–Dyson skew information.

     

Incompressible Functions,Relative-Error Extractors,and the Power of Nondeterministic Reductions

A circuit C compresses a function \({f : \{0,1\}^n\rightarrow \{0,1\}^m}\) if given an input \({x\in \{0,1\}^n}\), the circuit C can shrink x to a shorter ?-bit string x′ such that later, a computationally unbounded solver D will be able to compute f(x) based on x′. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size \({s=n^c}\). Motivated by cryptographic applications, we focus on average-case \({(\ell,\epsilon)}\) incompressibility, which guarantees that on a random input \({x\in \{0,1\}^n}\), for every size s circuit \({C:\{0,1\}^n\rightarrow \{0,1\}^{\ell}}\) and any unbounded solver D, the success probability \({\Pr_x[D(C(x))=f(x)]}\) is upper-bounded by \({2^{-m}+\epsilon}\). While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work, we present the following results:

1. (1)
   Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function \({f:\{0,1\}^n\rightarrow \{0,1\}}\) which is incompressible by size n ^c circuits with communication \({\ell=(1-o(1)) \cdot n}\) and error \({\epsilon=n^{-c}}\). Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14).
    
2. (2)
   We show that it is possible to achieve negligible error parameter \({\epsilon=n^{-\omega(1)}}\) for nonboolean functions. Specifically, assuming that E is hard for exponential size \({\Sigma_3}\)-circuits, we construct a nonboolean function \({f:\{0,1\}^n\rightarrow \{0,1\}^m}\) which is incompressible by size n ^c circuits with \({\ell=\Omega(n)}\) and extremely small \({\epsilon=n^{-c} \cdot 2^{-m}}\). Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest.
    
3. (3)
   We show that the task of constructing an incompressible boolean function \({f:\{0,1\}^n\rightarrow \{0,1\}}\) with negligible error parameter \({\epsilon}\) cannot be achieved by “existing proof techniques”. Namely, nondeterministic reductions (or even \({\Sigma_i}\) reductions) cannot get \({\epsilon=n^{-\omega(1)}}\) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospect, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (STOC 08).
    

     

Parallel Multi-Block ADMM with <Emphasis Type="Italic">o</Emphasis>(1 / <Emphasis Type="Italic">k</Emphasis>) Convergence

This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints:

$$\begin{aligned} \text {minimize} ~~&f_1(\mathbf{x}_1) + \cdots + f_N(\mathbf{x}_N)\\ \text {subject to}~~&A_1 \mathbf{x}_1 ~+ \cdots + A_N\mathbf{x}_N =c,\\&\mathbf{x}_1\in {\mathcal {X}}_1,~\ldots , ~\mathbf{x}_N\in {\mathcal {X}}_N, \end{aligned}$$

where \(N \ge 2\), \(f_i\) are convex functions, \(A_i\) are matrices, and \({\mathcal {X}}_i\) are feasible sets for variable \(\mathbf{x}_i\). Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices \(A_i\) are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices \(A_i\)). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that \(\Vert {\mathbf {x}}^{k+1} - {\mathbf {x}}^k\Vert _M^2\) converges at a rate of o(1 / k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported.

     

Exact asymptotics of small deviations for a stationary Ornstein-Uhlenbeck process and some Gaussian diffusion processes in the L p -norm, 2 ≤ p ≤ ∞

We prove results on exact asymptotics of the probabilities

Formule di quadratura «chiuse» di tipo gaussiano: Tabulzione dei nodi e dei coefficienti

The coefficientsA _hi and the nodesx _mi for «closed” Gaussian-type quadrature formulae $$\int\limits_{ - 1}^1 {f(x)dx = \sum\limits_{h = 0}^{2_8 } {\sum\limits_{i = 0}^{m + 1} {A_{hi} f^{(h)} (x_{mi} ) + R\left[ {f(x)} \right]} } } $$ withx _m0 =?1,x _{m, m+1} =1 andR[f(x)]=0 iff(x) is a polinomial of degree at most2m(s+1)+2(2s+1)?1, have been tabulated for the cases: $$\left\{ \begin{gathered} s = 1,2 \hfill \\ m = 2,3,4,5 \hfill \\ \end{gathered} \right.$$ .     

Sulle formule di quadratura di Tschebyscheff

For the Tschebyscheff quadrature formula: $$\int\limits_{ - 1}^1 {\left( {1 - x^2 } \right)^{\lambda - 1/2} f(x) dx} = K_n \sum\limits_{k = 1}^n {f(x_{n,k} )} + R_n (f), \lambda > 0$$ it is shown that the degre,N, of exactness is bounded by: $$N \leqslant C(\lambda )n^{1/(2\lambda + 1)} $$ whereC(λ) is a convenient function of λ. For λ=1 the complete solution of Tschebyscheff's problem is given.     

A Nyström method for a class of Fredholm integral equations on the real semiaxis

A class of Fredholm integral equations of the second kind, with respect to the exponential weight function \(w(x)=\exp (-(x^{-\alpha }+x^\beta ))\), \(\alpha >0\), \(\beta >1\), on \((0,+\infty )\), is considered. The kernel k(x, y) and the function g(x) in such kind of equations,

$$\begin{aligned} f(x)-\mu \int _0^{+\infty }k(x,y)f(y)w(y)\mathrm {d}y =g(x),\quad x\in (0,+\infty ), \end{aligned}$$

can grow exponentially with respect to their arguments, when they approach to \(0^+\) and/or \(+\infty \). We propose a simple and suitable Nyström-type method for solving these equations. The study of the stability and the convergence of this numerical method in based on our results on weighted polynomial approximation and “truncated” Gaussian rules, recently published in Mastroianni and Notarangelo (Acta Math Hung, 142:167–198, 2014), and Mastroianni, Milovanovi? and Notarangelo (IMA J Numer Anal 34:1654–1685, 2014) respectively. Moreover, we prove a priori error estimates and give some numerical examples. A comparison with other Nyström methods is also included.

     

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:

Non-determinism in Gödel’s System <Emphasis Type="Italic">T</Emphasis>

The calculus T^? is a successor-free version of Gödel’s T. It is well known that a number of important complexity classes, like e.g. the classes logspace, \(\textsc{p}\), \(\textsc{linspace}\), \(\textsc{etime}\) and \(\textsc{pspace}\), are captured by natural fragments of T^? and related calculi. We introduce the calculus T^∽, which is a non-deterministic variant of T^?, and compare the computational power of T^∽ and T^?. First, we provide a denotational semantics for T^∽ and prove this semantics to be adequate. Furthermore, we prove that \(\textsc{linspace}\subseteq \mathcal {G}^{\backsim }_{0} \subseteq \textsc{linspace}\) and \(\textsc{etime}\subseteq \mathcal {G}^{\backsim }_{1} \subseteq \textsc{pspace}\) where \(\mathcal {G}^{\backsim }_{0}\) and \(\mathcal {G}^{\backsim }_{1}\) are classes of problems decidable by certain fragments of T^∽. (It is proved elsewhere that the corresponding fragments of T^? equal respectively \(\textsc{linspace}\) and \(\textsc{etime}\).) Finally, we show a way to interpret T^∽ in T^?.     

Alcuni problemi relativi alle formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the form $$\int\limits_{ - 1}^1 {(1 - x)^a (1 + x)^\beta f(x)dx = \sum\limits_{i = 0}^{r - 1} {[A_i f^{(i)} ( - 1) + B_i f^{(i)} (1)] + K_n (\alpha ,\beta ;r)\sum\limits_{i = 1}^n {f(x_{n,i} ),} } } $$ (α>?1, β>?1), with realA _i ,B _i ,K _n and real nodesx _n,i in (?1,1), valid for prolynomials of degree ≤2n+2r?1. In the first part we prove that there is validity for polynomials exactly of degree2n+2r?1 if and only if α=β=?1/2 andr=0 orr=1. In the second part we consider the problem of the existence of the formula $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(x)dx = A_n f( - 1) + B_n f(1) + C\sum\limits_{i = 1}^n {f(x_{n,i} )} }$$ for polynomials of degree ≤n+2. Some numerical results are given when λ=1/2.