Class Of Generalized Functions Research Articles

A Nitsche-type method for Helmholtz equation with an embedded acoustically permeable interface

We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche’s method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function Z that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete Gårding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.

Some inequalities for chord power integrals of parallelotopes

We prove some geometric inequalities for pth-order chord power integrals \({\mathcal I}_p(P_d),\,1 \le p \le d,\) of d-parallelotopes \(P_d\) with positive volume \(V_d(P_d)\). First, we derive upper and lower bounds of the ratio \({\mathcal I}_p(P_d)/V_d^2(P_d)\) which are attained by a d-cuboid \(C_d\) with the same volume resp. the same mean breadth as \(P_d\). Second, we apply the device of Schur-convexity to obtain bounds of \({\mathcal I}_p(C_d)/V_d^2(C_d)\) which are attained by a d-cube with the same volume resp. the same mean breadth as \(C_d\). Most of these inequalities are shown for a more general class of ovoid functionals containing, as by-product, a Pfiefer-type inequality for d-parallelotopes.

Gaussian Process Planning with Lipschitz Continuous Reward Functions: Towards Unifying Bayesian Optimization, Active Learning, and Beyond

This paper presents a novel nonmyopic adaptive Gaussian process planning (GPP) framework endowed with a general class of Lipschitz continuous reward functions that can unify some active learning/sensing and Bayesian optimization criteria and offer practitioners some flexibility to specify their desired choices for defining new tasks/problems. In particular, it utilizes a principled Bayesian sequential decision problem framework for jointly and naturally optimizing the exploration-exploitation trade-off. In general, the resulting induced GPP policy cannot be derived exactly due to an uncountable set of candidate observations. A key contribution of our work here thus lies in exploiting the Lipschitz continuity of the reward functions to solve for a nonmyopic adaptive epsilon-optimal GPP (epsilon-GPP) policy. To plan in real time, we further propose an asymptotically optimal, branch-and-bound anytime variant of epsilon-GPP with performance guarantee. We empirically demonstrate the effectiveness of our epsilon-GPP policy and its anytime variant in Bayesian optimization and an energy harvesting task.

Implementation of fractional-order transfer functions in the viewpoint of the required fractional-order capacitors

ABSTRACTIn this paper, based on the concept of restricted difference basis (RDB) for a natural number, the general structures of the pseudo state-space realisations of some classes of commensurate fractional-order transfer functions are obtained. It is shown that the number of needed fractional-order capacitors for implementing a transfer function belonging to each of these classes depends on the commensurate order and also the RDB based on which the pseudo state-space realisation of the considered transfer function is obtained. In this regard, three general classes of commensurate fractional-order transfer functions are introduced which all can be implemented by the same number of fractional-order capacitors. Also, four classes of fractional-order transfer functions of commensurate order are introduced which all can be implemented by only one fractional-order capacitor of order . Finally, as a more general case, four classes of fractional-order transfer functions of commensurate order are studied in aspect of the number and the order of the fractional-order capacitors needed for their implementations. It is proved that these classes of commensurate fractional-order transfer functions can be implemented by r − 1 number of fractional-order capacitors of order .

Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices

Based on the multivariate saddle point method we study the asymptotic behavior of the characteristic polynomials associated to Wishart type random matrices that are formed as products consisting of independent standard complex Gaussian and a truncated Haar distributed unitary random matrix. These polynomials form a general class of hypergeometric functions of type 2 F r . We describe the oscillatory behavior on the asymptotic interval of zeros by means of formulae of Plancherel–Rotach type and subsequently use it to obtain the limiting distribution of the suitably rescaled zeros. Moreover, we show that the asymptotic zero distribution lies in the class of Raney distributions and by introducing appropriate coordinates elementary and explicit characterizations are derived for the densities as well as for the distribution functions.

Partially linear additive quantile regression in ultra-high dimension

We consider a flexible semiparametric quantile regression model for analyzing high dimensional heterogeneous data. This model has several appealing features: (1) By considering different conditional quantiles, we may obtain a more complete picture of the conditional distribution of a response variable given high dimensional covariates. (2) The sparsity level is allowed to be different at different quantile levels. (3) The partially linear additive structure accommodates nonlinearity and circumvents the curse of dimensionality. (4) It is naturally robust to heavy-tailed distributions. In this paper, we approximate the nonlinear components using B-spline basis functions. We first study estimation under this model when the nonzero components are known in advance and the number of covariates in the linear part diverges. We then investigate a nonconvex penalized estimator for simultaneous variable selection and estimation. We derive its oracle property for a general class of nonconvex penalty functions in the presence of ultra-high dimensional covariates under relaxed conditions. To tackle the challenges of nonsmooth loss function, nonconvex penalty function and the presence of nonlinear components, we combine a recently developed convex-differencing method with modern empirical process techniques. Monte Carlo simulations and an application to a microarray study demonstrate the effectiveness of the proposed method. We also discuss how the method for a single quantile of interest can be extended to simultaneous variable selection and estimation at multiple quantiles.

Rate of convergence of the Bézier variant of an operator involving Laguerre polynomials

In the present paper, we consider the Bézier variant of an operator involving Laguerre polynomials of degree k, with α≥1, for bounded functions f defined on the interval [0,1]. In particular, by using the Chanturia modulus of variation, we estimate the rate of pointwise convergence of (Pn,αf)(x,t) at those points x∈(0,1) at which the one‐sided limits f(x+),f(x−) exist. To prove our main result, we have used some methods and techniques of probability theory. Our result extends and generalizes the very recent results of very recent results of the authors to more general classes of functions. Copyright © 2016 John Wiley & Sons, Ltd.

Triple variational principles for self-adjoint operator functions

For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find conditions which imply that a point is in the resolvent set. For norm resolvent continuous operator functions we show that the variational inequality becomes an equality.

Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

In a continuous-time economy, we investigate the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation between the two risky assets. The portfolio selection that is robust to the uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation. Thus, it becomes a maximin problem. We solve the problem under the Black--Scholes model for risky assets with an ambiguous correlation using the theory of $G$-Brownian motions. We then extend the problem to stochastic volatility models for risky assets with an ambiguous correlation between risky asset returns. An asymptotic closed-form solution is derived for a general class of utility functions, including constant relative risk aversion and constant absolute risk aversion utilities, when stochastic volatilities are fast mean reverting. We propose a practical trading strategy that combines information from the option implied volatility surfaces of risky assets through the ambiguous correlation.

Periodic Oscillations for Nonmonotonic Smooth Negative Feedback Circuits

Negative feedback circuits are a recurrent motif in regulatory biological networks, strongly linked to the emergence of oscillatory behavior. The theoretical analysis of the existence of oscillations is a difficult problem and typically involves many constraints on the monotonicity of the activity functions. Here, we study the occurrence of periodic solutions in an n-dimensional class of nega- tive feedback systems defined by smooth vector fields with a window of not necessarily monotonic activity. Our method consists in circumscribing the smooth system by two piecewise linear ones, each admitting a periodic solution. It can then be shown that the smooth negative feedback system also has a periodic orbit, inscribed in the topological solid torus constructed from the two piecewise linear orbits. The interest of our approach lies in first, adopting a general class of functions, with a nonmonotonicity window, which permits a better fitting between theoretical models and experimen- tal data, and second, establishing a more accurate location for the periodic solution, which is useful for computational purposes in high dimensions. As an illustration, a model for the “repressilator” synthetic system is analyzed and compared to real data, and shown to admit a periodic orbit, for a range of activity functions.

Hybrid Fuzzy Jordan Network for Robust and Efficient Intrusion Detection System

Objective: The main objective of this paper is to improve the intrusion detection accuracy of neural networks by hybridized with Jordan network, applying novel Lyapunov function and changing input values as fuzzy values using fuzzy logic. Methods: In this research, the intrusion detection of NSL KDD dataset is carry out by neural network first . The performance of neural networks mainly depends on the parameters like number of hidden layer, no of node in the hidden layer and the no of epoch. Selecting the proper parameters values and weight initialization are the main difficulty in neural network. To overcome this issue a hybrid network by combining neural network with Jordan network is proposed which is highly sensitive to weight convergence. In hybrid network Lyapunov function is used to achieve the stability of equilibrium between two networks. But in this work a novel Lyapunov function is proposed to achieve global robust stability in hybrid network. A novel Lyapunov function uses a class of general activation functions which are not to be differentiable, bounded or monotonically nondecreasing. A set of criteria are derived to guarantee the existence, uniqueness and global robust stability of the equilibrium of hybrid networks with time delays. Then the learning ability of hybrid network is improved by using fuzzy logic. The hybrid network improved by a novel Lyapunov and fuzzy logic is called as Improved Fuzzy Hybrid Jordan network and Artificial Neural Network (IHFJANN). Findings: Artificial Neural Network (ANN), Hybrid Jordan network and Artificial Neural Network (HJANN) and Improved Hybrid Fuzzy Jordan network and Artificial Neural Network (IHFJ ANN) are applied on NSL KDD training dataset to lean the type of available attacks. NSL training dataset contains 2500 instances and 38 attributes where as testing dataset contains 995 instances and 38 attributes. The learned model of three classifiers is used to predict the classes in test dataset. The performance measures are evaluated in terms of accuracy, precision, recall and f-measure values. Improvement: The classification accuracy of Artificial Neural Network (ANN), Hybrid Jordan network and Artificial Neural Network (HJANN) and Improved Hybrid Fuzzy Jordan network and Artificial Neural Network (IHFJANN) are 72%, 80% and 84% respectively. Accuracy is increased by 12% in IHFJANN than the ANN and 4% than the HJANN. The precision value of ANN, HJANN and IHFJANN are 0.46, 0.54 and 0.57 respectively. Precision value is increased by 0.9 in IHFJANN than the ANN and 0.3 than the HJANN. The recall value of ANN, HJANN and IHFJANN are 0.73, 0.81 and 0.84 respectively. Recall value is increased by 0.9 in IHFJANN than the ANN and 0.3 than the HJANN. The F-measure value of ANN, HJANN and IHFJANN are 0.58, 0.65 and 0.68 respectively. F-measure value is increased by 0.10 in IHFJANN than the ANN and 0.3 than the HJANN. The results proved that the proposed IHFJANN provides better performance than ANN and HJANN.

Fractal Polynomials and Maps in Approximation of Continuous Functions

ABSTRACTThe primary goal of this article is to establish some approximation properties of fractal functions. More specifically, we establish that a monotone continuous real-valued function can be uniformly approximated with a monotone fractal polynomial, which in addition agrees with the function on an arbitrarily given finite set of points. Furthermore, the simultaneous approximation and \\mboxinterpolation which is norm-preserving property of fractal polynomials is established. In the final part of the article, we establish differentiability of a more general class of fractal functions. It is shown that these smooth fractal functions and their derivatives are good approximants for the original function and its \\mboxderivatives.

An Expressive Mechanism for Auctions on the Web

Auctions are widely used on the Web. Applications range from sponsored search to platforms such as eBay. In these and in many other applications the auctions in use are single-/multi-item auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is the limited expressiveness that they offer to the bidders. The General Auction Mechanism (GAM) of Aggarwal et al. [2009] takes a first step toward addressing the problem of limited expressiveness by computing a bidder optimal, envy-free outcome for linear utility functions with identical slopes and a single discontinuity per bidder-item pair. We show that in many practical situations this does not suffice to adequately model the preferences of the bidders, and we overcome this problem by presenting the first mechanism for piecewise linear utility functions with nonidentical slopes and multiple discontinuities . Our mechanism runs in polynomial time. Like GAM it is incentive compatible for inputs that fulfill a certain nondegeneracy assumption, but our requirement is more general than the requirement of GAM. For discontinuous utility functions that are nondegenerate as well as for continuous utility functions the outcome of our mechanism is a competitive equilibrium . We also show how our mechanism can be used to compute approximately bidder optimal, envy-free outcomes for a general class of continuous utility functions via piecewise linear approximation. Finally, we prove hardness results for even more expressive settings.

Teamwork Efficiency and Company Size

AbstractWe study how ownership structure and management objectives interact in determining the company size without assuming information constraints or any explicit costs of management. In symmetric agent economies, the optimal company size balances the returns to scale of the production function and the returns to collaboration efficiency. For a general class of payoff functions, we characterize the optimal company size, and we compare the optimal company size across different managerial objectives. We demonstrate the restrictiveness of common assumptions on effort aggregation (e.g., constant elasticity of effort substitution), and we show that common intuition (e.g., that corporate companies are more efficient and therefore will be larger than equal-share partnerships) might not hold in general.

Fast tensor method for summation of long‐range potentials on 3D lattices with defects

SummaryIn this paper, we present a method for fast summation of long‐range potentials on 3D lattices with multiple defects and having non‐rectangular geometries, based on rank‐structured tensor representations. This is a significant generalization of our recent technique for the grid‐based summation of electrostatic potentials on the rectangular L × L × L lattices by using the canonical tensor decompositions and yielding the O(L) computational complexity instead of O(L3) by traditional approaches. The resulting lattice sum is calculated as a Tucker or canonical representation whose directional vectors are assembled by the 1D summation of the generating vectors for the shifted reference tensor, once precomputed on large N × N × N representation grid in a 3D bounding box. The tensor numerical treatment of defects is performed in an algebraic way by simple summation of tensors in the canonical or Tucker formats. To diminish the considerable increase in the tensor rank of the resulting potential sum, the ϵ‐rank reduction procedure is applied based on the generalized reduced higher‐order SVD scheme. For the reduced higher‐order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices. The required storage scales linearly in the 1D grid‐size, O(N), while the numerical cost is estimated by O(NL). The approach applies to a general class of kernel functions including those for the Newton, Slater, Yukawa, Lennard‐Jones, and dipole‐dipole interactions. Numerical tests confirm the efficiency of the presented tensor summation method; we demonstrate that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in seconds in Matlab implementation. The tensor approach is advantageous in further functional calculus with the lattice potential sums represented on a 3D grid, like integration or differentiation, using tensor arithmetics of 1D complexity. Copyright © 2015 John Wiley & Sons, Ltd.

Path integral solution for nonlinear systems under parametric Poissonian white noise input

In this paper the problem of the response evaluation in terms of probability density function of nonlinear systems under parametric Poisson white noise is addressed. Specifically, extension of the Path Integral method to this kind of systems is introduced. Such systems exhibit a jump at each impulse occurrence, whose value is obtained in closed form considering two general classes of nonlinear multiplicative functions. Relying on the obtained closed form relation liking the impulses amplitude distribution and the corresponding jump response of the system, the Path Integral method is extended to deal with systems driven by parametric Poissonian white noise. Several numerical applications are performed to show the accuracy of the results and comparison with pertinent Monte Carlo simulation data assesses the reliability of the proposed procedure.

Some fixed point theorems in Menger probabilistic metric-like spaces

In this manuscript we extend very recent fixed point theorems in the setting of Menger spaces in three senses: on the one hand, we introduce the notion of Menger probabilistic metric-like space by avoiding a non-necessary constraint (at least, for our purposes) in the properties that define a Menger space; on the other hand, we consider a more general class of auxiliary functions in the contractivity condition; finally, we show that the function $t\rightarrow1/t-1$ (which appears in many fixed point theorems in the fuzzy context) can be replaced by more appropriate and general functions. We illustrate our main statements with an example in which previous results cannot be applied.

Direct evaluation of jumps for nonlinear systems under external and multiplicative impulses

In this paper the problem of the response evaluation of nonlinear systems under multiplicative impulsive input is treated. Such systems exhibit a jump at each impulse occurrence, whose value cannot be predicted through the classical differential calculus. In this context here the correct jump evaluation of nonlinear systems is obtained in closed form for two general classes of nonlinear multiplicative functions. Analysis has been performed to show the different typical behaviors of the response, which in some cases could diverge or converge to zero instantaneously, depending on the amplitude of the Dirac's delta.

On the singlet projector and the monodromy relation for psu(2, 2|4) spin chains and reduction to subsectors

As a step toward uncovering the relation between the weak and the strong coupling regimes of the $\mathcal{N}=4$ super Yang-Mills theory beyond the specral level, we have developed in a previous paper [arXiv:1410.8533] a novel group theoretic interpretation of the Wick contraction of the fields, which allowed us to compute a much more general class of three-point functions in the SU(2) sector, as in the case of strong coupling [arXiv:1312.3727], directly in terms of the determinant representation of the partial domain wall partition funciton. Furthermore, we derived a non-trivial identity for the three point functions with monodromy operators inserted, being the discrete counterpart of the global monodromy condition which played such a crucial role in the computation at strong coupling. In this companion paper, we shall extend our study to the entire ${\rm psu}(2,2|4)$ sector and obtain several important generalizations. They include in particular (i) the manifestly conformally covariant construction, from the basic principle, of the singlet-projection operator for performing the Wick contraction and (ii) the derivation of the monodromy relation for the case of the so-called "harmonic R-matrix", as well as for the usual fundamental R-matrtix. The former case, which is new and has features rather different from the latter, is expected to have important applications. We also describe how the form of the monodromy relation is modified as ${\rm psu}(2,2|4)$ is reduced to its subsectors.

Persistence Barcodes Versus Kolmogorov Signatures: Detecting Modes of One-Dimensional Signals

We investigate the problem of estimating the number of modes (i.e., local maxima) - a well known question in statistical inference - and we show how to do so without presmoothing the data. To this end, we modify the ideas of persistence barcodes by first relating persistence values in dimension one to distances (with respect to the supremum norm) to the sets of functions with a given number of modes, and subsequently working with norms different from the supremum norm. As a particular case we investigate the Kolmogorov norm. We argue that this modification has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples.