Simple Closed-form Solution Research Articles

Subpixel edge localization with reduced uncertainty by violating the Nyquist criterion

In this contribution, the extent to which the Nyquist criterion can be violated in optical imaging systems with a digital sensor, e.g., a digital microscope, is investigated. In detail, we analyze the subpixel uncertainty of the detected position of a step edge, the edge of a stripe with a varying width, and that of a periodic rectangular pattern for varying pixel pitches of the sensor, thus also in aliased conditions. The analysis includes the investigation of different algorithms of edge localization based on direct fitting or based on the derivative of the edge profile, such as the common centroid method. In addition to the systematic error of these algorithms, the influence of the photon noise (PN) is included in the investigation. A simplified closed form solution for the uncertainty of the edge position caused by the PN is derived. The presented results show that, in the vast majority of cases, the pixel pitch can exceed the Nyquist sampling distance by about 50% without an increase of the uncertainty of edge localization. This allows one to increase the field-of-view without increasing the resolution of the sensor and to decrease the size of the setup by reducing the magnification. Experimental results confirm the simulation results.

An adaptive quantization algorithm without side information for depth map coding

This paper presents a novel block-adaptive quantization scheme for efficient bit allocation without side information in depth map coding. Since the type of distortion in a depth map causes different effects in terms of the visual artifacts in a synthesized view, the proposed method adaptively assigns the number of bits according to the characteristics of the corresponding texture block. I have studied the details of the depth map and its rendered view distortion, modeled these analytically, and then proposed a new rate and distortion model for depth map coding. Finally, I derived a simple closed-form solution based on my proposed rate and distortion model, which determines the block-adaptive quantization parameter without any side information. Experimental results show that the proposed scheme can achieve coding gains of more than 0.6% and 1.4% for quarter- and full-resolution depth maps, respectively, in a multi-view-plus-depth 3D system.

Analytical solutions of stresses and displacements for deep circular tunnels with liners in saturated ground

To investigate the displacement and stress distributions for deep circular tunnels with liners in saturated ground, an analytical model is proposed. For a deep tunnel with drainage conditions, plane strain conditions at any cross-section of the tunnel and the elastic regime of the linear elasticity for the remaining liner are assumed, while the ground is assumed to be linearly elastic and perfectly plastic with a failure surface de?ned by the Mohr-Coulomb criterion. The post-yield behavior of the ground follows the non-associated flow rule defined by the dilation angle. To solve the proposed problem, two procedures are presented. An axisymmetric model for a deep circular tunnel with a steady-state seepage condition is considered, and then a simple closed-form analytical solution is obtained with a common theoretical framework for the boundary conditions of a constant total head along the tunnel circumference. Assuming that certain ground displacements along the tunnel circumference have occurred before the installation of the liner, analytical solutions of stresses and displacements are derived with particular emphasis on the seepage and the stress release effect induced by tunnelling. The proposed analytical model is validated by numerical simulation.

Simplified Modeling of Rectangular Concrete Cross-Sections Confined by External FRP Wrapping

The goal of this research project is to model the effect of confinement by means of fiber reinforced polymer (FRP) externally bonded wrapping, hence to provide a simplified closed form solution to determine directly the ultimate confined concrete strength. Common cross-section shapes for reinforced concrete (RC) columns are considered herein, namely square and rectangular. The simplified model is derived from a more refined iterative confinement model proposed by the same authors to evaluate the entire stress-strain relationship of confined concrete. Based on a detailed analysis of the stress state through Mohr’s circle, a simplified closed form solution is proposed to account for the non-uniformly confined concrete performance exhibited in non-axisymmetric sections. The non-uniform confining stress field exhibited in such cross-sections is explicitly considered by means of the mean value integral of the pointwise variable stress state over the cross-section. The key aspect of the proposed methodology is the evaluation of the effective equivalent pressure to be inserted in any triaxial confinement model, to account for the peculiarities of square and rectangular cross-sections. Experimental data, available in the literature and representative of a wide stock of applications, were compared to the results of the theoretical simplified model to validate the proposed approach, and satisfactory results were found.

Analytical approach for transverse vibration analysis of castellated beams

This paper presents an analytical study on the dynamic characteristics of castellated beams. The study focuses on the effect of web shear on the free vibration frequencies of castellated beams. By using the Hamilton's principle, a simple closed-form solution for determining the free vibration frequencies of simply supported castellated beams is developed. The results show that the shear effect on the free vibration frequencies increases with the cross-sectional area and distance between the centroids of the two tee sections of castellated beams, but decreases with respect to increasing web thickness or increasing beam length. The shear effect is also found to be greater in higher vibration modes.

A Closed-Form, Consistent and Robust Solution to Uncalibrated Photometric Stereo Via Local Diffuse Reflectance Maxima

Images of an object under different illumination are known to provide strong cues about the object surface. A mathematical formalization of how to recover the normal map of such a surface leads to the so-called uncalibrated photometric stereo problem. In the simplest instance, this problem can be reduced to the task of identifying only three parameters: the so-called generalized bas-relief (GBR) ambiguity. The challenge is to find additional general assumptions about the object, that identify these parameters uniquely. Current approaches are not consistent, i.e., they provide different solutions when run multiple times on the same data. To address this limitation, we propose exploiting local diffuse reflectance (LDR) maxima, i.e., points in the scene where the normal vector is parallel to the illumination direction (see Fig. 1). We demonstrate several noteworthy properties of these maxima: a closed-form solution, computational efficiency and GBR consistency. An LDR maximum yields a simple closed-form solution corresponding to a semi-circle in the GBR parameters space (see Fig. 2); because as few as two diffuse maxima in different images identify a unique solution, the identification of the GBR parameters can be achieved very efficiently; finally, the algorithm is consistent as it always returns the same solution given the same data. Our algorithm is also remarkably robust: It can obtain an accurate estimate of the GBR parameters even with extremely high levels of outliers in the detected maxima (up to 80 % of the observations). The method is validated on real data and achieves state-of-the-art results.

Closed-Form Solution of Visual-Inertial Structure from Motion

This paper investigates the visual-inertial structure from motion problem. A simple closed form solution to this problem is introduced. Special attention is devoted to identify the conditions under which the problem has a finite number of solutions. Specifically, it is shown that the problem can have a unique solution, two distinct solutions and infinite solutions depending on the trajectory, on the number of point-features and on their layout and on the number of camera images. The investigation is also performed in the case when the inertial data are biased, showing that, in this latter case, more images and more restrictive conditions on the trajectory are required for the problem resolvability.

Optimal limit order execution in a simple model for market microstructure dynamics

Market participants who have the task of acquiring a certain position in a listed security at a predetermined price on behalf of a third party with no time urgency, ie, to fill a perpetual limit order, can optimize the profitability of their trading strategy in order to accomplish this task.We study the statistical properties of the profit distribution of a particular market-making strategy: one that increments the inventory as the underlying price approaches the limit order price So and locks in profits by gradually liquidating the inventory as the market drifts away from So. We do so by adopting a simple model of market microstructure, in which an unobservable continuous stochastic process, the microprice, drives the dynamics of limit and market orders. In this model, the arrival of market orders and updates of the limit order book are determined by the microprice crossing a discrete set of n equidistant levels between the price ticks. Assuming normal dynamics for the microprice and adopting a standard mean-variance framework, we are able to derive a remarkably simple closed-form solution for the optimal inventory profile: the cumulative amount held when the market price is Si is inversely proportional to Si - So, the distance in price terms from the limit order price. Finally, we show that n represents a sort of microvolatility of the market that is distinct from the diffusive volatility of the microprice and is a measure of the intensity of the bid-ask bounce.

Improvement of TDOA Position Fixing Using the Likelihood Curvature

A conventional multilateration is a two-step process where, in the first step the time difference of arrivals (TDOA) of a signal at multiple sensors are estimated, and in the second step, these TDOAs are used in some position fixing technique to estimate the location of an emitter. Several estimators have been proposed over the years for the estimation of the TDOAs. Many techniques have been proposed for position fixing as well. Much of the research on position fixing has been focused on obtaining a simplified closed form solution. For the unknown deterministic signal model, Stein had derived the maximum-likelihood estimator (MLE) for the TDOA between two sensors, which is the peak location of the cross-correlation function. Since the asymptotic variance of an MLE approaches the Cramer-Rao lower bound (CRLB), which is the inverse of the negative of the expected value of the curvature of the log-likelihood function, using this as a motivation, we propose a weighted least squares type position fixing technique where the weights are computed from the curvature of the log-likelihood function.

Worst-Case Complexity of Smoothing Quadratic Regularization Methods for Non-Lipschitzian Optimization

In this paper, we propose a smoothing quadratic regularization (SQR) algorithm for solving a class of nonsmooth nonconvex, perhaps even non-Lipschitzian minimization problems, which has wide applications in statistics and sparse reconstruction. The proposed SQR algorithm is a first order method. At each iteration, the SQR algorithm solves a strongly convex quadratic minimization problem with a diagonal Hessian matrix, which has a simple closed-form solution that is inexpensive to calculate. We show that the worst-case complexity of reaching an $\epsilon$ scaled stationary point is $O(\epsilon^{-2})$. Moreover, if the objective function is locally Lipschitz continuous, the SQR algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an $\epsilon$ Clarke stationary point in at most $O(\epsilon^{-3})$ iterations.

BLACK–SCHOLES REPRESENTATION FOR ASIAN OPTIONS

Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond BT with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.

New Properties of the Old Present Value Operator

Advances in computing technology have greatly enhanced methods for numerical calculations of present value and related measures such as duration and convexity. Nevertheless, closed form solutions continue to play an important role both in the classroom and in the real world. For example, it is well known that if r is the rate of discount and if C1 denotes the value in period 1 for a cash flow that grows at constant percentage rate, g, then the present value of the future cash flow can be represented as C1 / (r – g). Yet how many students or practitioners, and dare we ask how many finance professors, are aware that the duration of a perpetual cash flow that grows at a uniform geometric rate can be represented as (1 r) / (r –g) ? For that matter, how widely is it known that a simple closed form solution exists for the present value of a cash flow that exhibits cyclical variation over time or a cash flow that grows by a constant dollar amount each period rather than by a constant percentage amount? The objective of this paper is to demonstrate that these results, and countless others, can be derived from one simple but previously under developed property of the traditional present value operator.

Load transfer in adhesive double-sided patch joints

In this work, the application of adhesively bonded joints to connect two structural elements with a double-sided patch is studied. On the basis of the shear lag model, a simple closed-form solution was obtained. The analytical solutions can be used to predict the shear stress in the adhesive and the load transfer between the structural elements and the external patches. The load and shear stress distributions in the adhesively bonded region are presented. For verification of the analytical model, finite element analyses were employed to calculate the load transfer and shear stress for the double-sided patch joint under static tensile loadings. Good agreement was found between the theoretical predictions and numerical results. To obtain a better understanding of the joints, the effects of adhesive thickness, adhesive shear modulus and patch Young's modulus on the load transfer and shear stress distributions were investigated. The results show that the maximum shear stress occurs at the edge of the adhesive. The maximum value of the shear stress increases as the adhesive shear modulus and patch Young's modulus increase and as the adhesive thickness decreases. A more gradual load transfer can be achieved by increasing the adhesive thickness and decreasing the adhesive shear modulus. The simple analytical solution presented in this paper has the advantages of avoiding the numerical difficulties and giving explicit relationship between the stress state and joint parameters. Moreover, from the designer's point of view a closed-form and easy-to-use solution is preferred.

Lateral-Torsional Buckling of Partially Composite Horizontally Layered or Sandwich-Type Beams under Uniform Moment

This paper is devoted to the analytical and numerical modeling of the lateral-torsional stability of horizontally layered composite beams. Composite beams are classified as horizontally layered beams with interlayer slip or sandwich beams with a weak shear core. The governing differential equations of the out-of-plane behavior of horizontally layered composite beams are supported by variational arguments. In the theoretical analysis, a distinction is made between the influence of the shear connection at the interface with respect to the in-plane or transversal deformations and to the out-of-plane or lateral deformations, respectively. Some engineering results are presented for a partially composite beam under pure bending moment. In the case of noncomposite in-plane action (orthotropic connection), a simple closed-form solution is derived for the lateral-torsional buckling moment, and it is shown that the exact dimensionless buckling moment depends only on two structural parameters for beams composed of two identical subelements. The results are analogous to those obtained for the in-plane buckling of partially composite or sandwich-type beams, where the buckling moment increases with the stiffness of the shear connection. Prandtl’s valid solution for lateral-torsional buckling of ordinary beams is also found for composite beams in the case of noncomposite action in both the transversal and lateral directions. A generalization of Prandtl’s valid solution for composite beams with partial composite action in the lateral direction and noncomposite action in the transversal direction is derived. It is shown that the lateral-torsional buckling formulas are strongly affected by the kinematics of the connected shear layer. Also, the lateral-torsional buckling of partially composite beams with both in-plane and out-of-plane slip behavior is analyzed using the Rayleigh-Ritz method. This mathematical problem leads to a system of differential equations with nonuniform coefficients. An approximated solution is derived for the isotropic connection with isotropic noncomposite actions, whereas an exact solution is presented for the orthotropic connection with noncomposite in-plane action. Finally, the Rayleigh-Ritz approach is compared with some numerical results associated with the exact resolution of the differential equations with nonuniform coefficients. The Rayleigh-Ritz approach appears to be efficient to capture the main phenomena, including the nonmonotonic dependence of the buckling load to the connection parameter.

Similitude study on bending stiffness and behavior of composite tubes

A procedure to design a composite tube that matches the flexural stiffness, load-carrying capacity, and energy absorption of an aluminum tube while subjected to bending load is of interest. The large deformation and energy absorption requirements are fulfilled through the progressive failure of the plies of suitable fiber orientations across the thickness of the tube. The layer wise nature of the composite tubes and cylindrical anisotropy, however, has impeded obtaining a simple closed-form solution for calculating the flexural stiffness. Available analytical solutions are mainly a complex set of equations, which should be solved simultaneously. In addition, the boundary and interface conditions between the adjacent plies must be satisfied. This article presents a straightforward simulation technique for this purpose. First, the tube is correlated to a corresponding composite sandwich panel, the flexural stiffness of which is obtained by the classical laminate theory. Then an analogous aluminum sandwich panel is designed and is correlated to a so-called ‘equivalent’ aluminum tube. The bending stiffness of the composite tube is shown to be the same as that of its equivalent aluminum tube. As a result, a complex problem of cylindrical anisotropy is mapped into a Cartesian coordinate system and solved via the classical laminate theory. The accuracy of the technique is verified by experimental work and analytical solutions. The agreement between the three methods is shown.

Noncoherent Orthogonal Distributed Space-Time Block Codes for Amplify-and-Forward Half-Duplex Cooperative Relay Channels

In this paper, the design of a simple orthogonal distributed space-time block code for a noncoherent amplify-and-forward (AF) half-duplex cooperative relay system is proposed by utilizing a pair of coprime phase-shift keying (PSK) constellations and the Alamouti coding scheme. It is shown that one significant advantage of such design is that, in a noise-free case, for any given nonzero received signals, the channel coefficients and the transmitted signals can be blindly and uniquely determined with a simple closed-form solution. The other significant advantage is that, in a noise case, the least square error (LSE) receiver also has a simple closed-form solution to jointly estimate the channel and the transmitted signals. Simulation results show that our proposed scheme obtains a significant gain of signal-to-noise ratio over the training scheme.

A kinematics study of variable lead axisymmetric forward spiral extrusion

Variable lead axi-symmetric forward spiral extrusion (VLAFSE) allows far more efficient strain accumulation than constant lead counterpart AFSE. It eliminates the rigid body rotation of the sample in an AFSE die which only causes frictional loss and has no contribution to the deformation. Based on a proposed velocity field, the kinematic formulation for VLAFSE is presented, utilized to predict strain components in the deformation zone and compared with experimental measurements. A simple closed form solution of the VLAFSE problem is presented which describes the flow of material inside the extrusion die. The results confirm that the effective strain of the process accumulates non-linearly in the both radial and extrusion directions.

Simple Closed-Form Solution for the Buckling of Moderately Thick Anisotropic Cylinders

Simple Closed-Form Solution for the Buckling of Moderately Thick Anisotropic Cylinders

The electrohydrodynamics of superimposed fluids subjected to a nonuniform transverse electric field

This study aims to investigate electrohydrodynamics of two superimposed fluids that are confined between a pair of two-dimensional flat plates and are exposed to a sinusoidal electric field in zero gravity. The goal is to identify the parameters that affect the flow structure and interface deformation using a simple closed form solution. The governing electrohydrodynamic equations are solved analytically for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electric and flow fields is presented and it is shown that the electric field induces sinusoidal electrical stresses at the interface, which lead to periodic convection cells. The parameters affecting the sense of flow circulation and strength are investigated and it is shown that the former depends on the relative magnitude of the electric permittivity and conductivity ratios while the latter is controlled by the relative thicknesses of the fluid layers and the ratio of the electric conductivities and viscosities of the fluids. The maximum flow strength is achieved at a relative thickness that is set by the competition between the electric and hydrodynamic effects. For small deformation, the distortion of the interface is examined using a normal stress balance at the interface, and it is shown that the degree of interface deformation scales with the square of the amplitude of the electric potential nonuniformity, while its wavenumber is twice that of the imposed potential nonuniformity. Furthermore, a zero-deformation curve is found, which delineates the region in the permittivity-conductivity space according to the sense of interface deformation. The results show that for certain ranges of fluid layer thicknesses and permittivity ratios, the interface will remain flat, despite the action of the nonuiform field.

A SURVEY ON AMERICAN OPTIONS: OLD APPROACHES AND NEW TRENDS

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European op- tion can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up un- til now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These ap- proaches typically provide numerical or approximate analytic methods to �nd the price and the boundary. Topics included in this survey are early approaches(treesnite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, an- alytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochas- tic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.