Geometric model for motion of curves specified by acceleration - Research Proposal Example

? Geometric Model for Motion of Curves Specified by Acceleration s of investigators affiliation Table of Contents 3 Introduction 4 Literature Survey 5 Objectives 9 Design 10 Research Plan 11 Anticipated Outcomes 12 References 14 Budget 16 abstract The analysis proposed for the Geometric Model for Motion of Curves Specified by Acceleration will seek to answer the research question: How is the motion of manifolds specified by acceleration described? Through a qualitative analysis, the examination will attempt to fulfill primary and secondary objectives through an in-depth review of pertinent literature as well as an examination of mathematical models that will assist in constructing and deconstructing geometrical models for motion curves. An extensive literature review will be conducted over a specified timeframe as the overall methodology and existing literature analyzed to provide the equation models that will serve as computational formats for the overall examination. This research will contribute to present knowledge by applying the existing equations to modern scenarios to determine whether the existing data remains relevant and provide new details pertinent to current trends. ???? ????? ??????? ????? ??????? ??????? ????? ????????? ????? ? [?????????] ?? ???? ????? ????: ??? ???? ?????? ??????? ????? ?????? ????? ?? ???? ????? ??????, ?????? ????? ?? ???? ??????? ?????? ??????? ?? ???? ??????? ?????? ?? ??? ?????? [?? ??? ?] ??? ?? ????? ??????? ??? ?????? ?? ???? ? [????????????] ????? ??????? ????? ???????. ????? ??? ????? ??????? ???? ??? [?] ????? [??????] ??? ??? ?????????? ??????? ???? ?????? ????? ?? ????? ???????? ????? ??? ????? ?????? ??????? ????? ????????. ????? ??? ??? ??? ????? ????? ? ????? ????????? ?????? ??? ?????????? ????? ?? ????? ?? ??? ???????? ?????? ???? ?????? ?????? ?????? ????? ?????? ??? ???????? ??????. Geometric Model for Motion of Curves Specified by Acceleration introduction A geometric model generally deals with the kinematics of a one dimensional manifold in a higher dimensional space. The model is specified by acceleration fields which are local or global functions of the intrinsic quantities of the manifold. This research intends to examine the evolution of one dimensional manifold embedded in the Euclidean space as it evolves under a stochastic flow of diffeomorphisms (1). Within the manifold, motion depends on the intrinsic invariants immersed in the space. During the course of this research, we will obtain the system of differential equations that governs the motion of the curve, keeping in mind that the processes driving the stochastic flows are chosen to be the most common class of Gaussian processes with stationary increments in time, which is the family of fractional Brownian motions with Hurst parameter (1). A family of random mappings is called a stochastic (Brownian) flow and is formulated as follows: “?st, 0 ?< s ?< t < ?, Rn into itself such that: ?st, for each s ?< t is a diffeomorphisms of Rn into itself. ?ut ? ?su = ?st, for all s ?< u ?< ?. ?tt is the identity map on Rn for all t. ?s1t1, ?s2t2, …, ?sntn are independent if s1 ?< t1 ?< s2 ?< t2 ?< … ?< sn ?< tn. ” (1 p.8) Using some applications to give geometric meanings to each solution to the governing system of (Partial Differential Equations) PDE,s (2) corresponding to the model length and local time investigated, this profile will also demonstrate how the geometric problem can be transformed to a fully nonlinear parabolic system of equations for the curvature, the position, and orientation. This research will also examine the primary curvature properties developed during the evolution of curves. Another facet of the study will explore the evolution of derive time equations using the Frenet frame. Further derive time equations will be determined regarding the intrinsic quantities satisfied by curves. The investigation will also propose a model using the solution of the evolution equation for the curvature and torsion and the Fundamental theorem for space curves to demonstrate the evolving space curve. A current and historic profile of relevant literature will be used in the illustration of the relevant properties of the geometric model for motion of curves specified by acceleration in the proposed qualitative analysis to answer the research question: How is the motion of manifolds specified by acceleration described? Literature Survey The study and analysis of the structure and motion of curves has traditionally been a field of significant interest such that there are numerous models and algorithms used to calculate the proportions and geometric physiology of curves (3). Examination of the mechanical structure of curves is a discipline based on the dynamics of how and why things move the way they do and identifies particles as specific points in Euclidean space represented by position vectors, cylindrical coordinates, spherical polar coordinates, or Cartesian coordinates (4). In mechanical terms, the definition of a particle indicated as “…a very useful, simplified model of ordinary objects, or bodies" (4 p.2). This definition is also mathematically represented in that the particle is considered to be: “…an object which is completely characterized by its position, ?r = xi + yi + zќ and velocity ?v = vxi + vyi + vzќ” (4 p.2). Since the point is identified in Euclidean space, the movement of this point can be established as a curve in Euclidean space, which can be calculated by the equation x = cos (t); y = sin (t); z = t (4 p.3), and the properties of such a curve include acceleration and velocity (4). The fundamental units used to measure movements are intervals in space, such as lengths, the quantities of inertia, or mass, comprising the different bodies, and the congruent intervals of time (5). Within the manifold, all measurements can be calculated in one of these units, whose representations are the meter, the kilogram, or the second (mks) and these units of measurement comprise the mks system used within science and engineering presently (5) and in this proposal. The first Newtonian principle of motion state that if the motion of a given body is not disturbed by external influences then that body moves with constant velocity, which can be represented by the equation r = r0 + v t (5). Newton’s second principle states that the change of motion of an object is directly proportional to the amount of force impressed upon it and the directional change is made in the direction of the straight line in which the force is impressed, which is expressed by the equation dp/dt = f, where motion or momentum is defined as p = mv and f represents the force exerted on the object, defined as f = ma (5). The third Newtonian law states that for every action there is an equal reaction or “the mutual actions of two bodies upon each other are always equal and directed to contrary parts” (5 p.53), which means that if we designate the “action” fab, according to Newton’s theory, fba would be the equal and opposite reaction, comprising the formula fab = fba (5). To phrase Newtonian theory more succinctly, “Newton's first law…asserts that space-time events can be organized into inertial reference frames where Newton's second law holds. It is not too hard to see that if there exists one inertial reference frame then there exists infinitely many others. In particular, given an inertial reference frame, defined by a choice of (t; x; y; z) one is free to (i) rotate to a new set of axes, (ii) shift the origin of space to a new location, (iii) shift the origin of time to a new instant, (iv) use a reference frame which is moving at constant relative velocity. This set of transformations defines the totality of all inertial reference frames. The fact that there are so many of them reflects the homogeneity and isotropy of space and the homogeneity of time which is fundamental to Newton’s mechanics” (8 p.5). In actuality, the motion of a particle enacts continuous change in its configuration, specifically, its position in time, or t (8). To this effect, mathematically, motion can be defined by selecting “three functions x (t), y (t), and z (t) such that the position of the particle (x; y; z) varies in time according to x = x (t); y = y (t); z = z (t); that is, ?r = ?r (t)” (8 p.3). The intrinsic invariants immersed within the space of the manifold will have an effect on the motion of the curve. Acceleration is definitively described as “the rate of change of velocity with time” (5 p.21) and the dimensions of an acceleration are [L]/ [T2] (5 p.14), where [L] represents length and [T] represents the interval of time or ? = ?v/?t (5 p.21). In this equation, ? represents the body's acceleration at time t, and ?v is the adjustment in velocity of the body between times t and t + ?t (5). However, since there are a multitude of factors that can affect the dimensions of the curve, calculating the exact proportions can be a difficult task. As noted by Tazawa, “integrating the curvature over a curve or constructing a curve with assigned curvature can be very difficult even in the simplest cases” (6 p.2). These calculations can become even more difficult in cases where the acceleration is inconsistent or the rate of acceleration is constantly fluctuating. As defined by the theoretical principles of Brownian movement, suspended particles are perpetually and randomly bombarded from all directions by the thermal molecular motions of the molecules surrounding them and, in the case of very small particles, the hits from one side will be stronger than those from other side, causing it to jump, which is what creates Brownian motion (7). As indicated by Le Jan, “…stochastic flows driven by smooth Brownian vector fields on a compact manifold define flows of diffeomorphisms” and from this we can determine that “flows of random coalescing maps or flows of random transition probabilities can arise from simple stochastic differential equations when Ito’s theory of strong solutions ceases to apply” (8 p.649). In these instances, the relative stochastic flows can be interpreted as having infinitely divisible limits of products of i.i.d. (independent and identically distributed) random diffeomorphisms, and the theory is similar to the theory of Brownian motion on Lie groups (8), which were originally introduced as a tool to solve or simplify ordinary and partial differential equations using the Galois model to solve algebraic equations of two, three, and four degree, and to show that the general polynomial equation of degree greater than four could not be solved by radicals (9). under an isotropic and volume preserving Brownian flow, the primary result will be a stochastic evolution equation for the Lipschitz-Killing curvatures of a erratically evolving manifold, which is an important consequence of that is an explicit, simple expression for their expected values as a function of time (1). The Hurst Parameter as defined by the self-similar process, represented by H, can be estimated using: 1) The time domain analysis R/S statistics, 2) An analysis of the variances of the aggregated processes, X(m) variance time plots, 3) A periodogram-based analysis in the frequency-domain, such as Whittles Maximum Likelihood Estimator (MLE) (10). The most common definition for the self-similar process in the Hurst model is: “X = (Xt? - ? < t < ?) is by means of its distribution true if (X?t) and ?H (Xt) have identical infinite-dimensional distributions for all ? > 0 then X is self-similar with parameter H” (10 p.1). However, when using the Hurst models, both the R/S and the periodogram-based analyses should be applied to the original and aggregated data sets to filter out some effects of the process behaviors at higher frequencies (10). Rose further illustrates numerous models of calculations based on the Hurst parameter (10). When Frenet defined his moving frame and his special equations, these elucidations have come to developed an important role in mechanics and kinematics as well as in differential geometry, the geometry of special relativity, and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold introduced and some classical differential geometry topics have been treated by since developed by researchers (11). As explained by Turgut and Yilmaz, “Minkowski space-time E41 is an Euclidean space E4 provided with the stan-dart flat metric given by g = ?dx21 + dx22 + dx23 + dx24 where (x1, x2, x3, x4) is a rectangular coordinate system in E41” (11 p.794), which directly correlates with the Frenet apparatus of an involute-evolute curve couple in E41 (11). This is better expressed by the theorem: “Let ? and ? be unit speed space-like curves and ? be an evolute of ?. The Frenet apparatus of ? ({T?, N?, B?, E?, ??, ??, ??}) can be formed according to Frenet apparatus of ? ({T,N,B,E, ?, ?, ?})” (11 p.797). Turgut and Yilzman exemplify and prove this theorem applied as well as a special characterization of an involute-evolute curve couple in E41 (11). Although it was previously commonly theorized that the Lyapunov exponents could explain the reappearance of the second primary form at any point of a manifold, evolving randomly under a stochastic flow, it was later discounted and new theories emerged which stipulated that the finite time actions of the geometric evolution of the flow may be better studied by other flow characteristics (1). Further studies, as indicated by Vadlamani, regarding the “unfolding of the symmetric polynomials of the principle curvatures, including the mean and the Gaussian curvature of an (n?1)-dimensional manifold M embedded in Rn, evolving under an isotropic and volume preserving flow on Rn” allowed “an Ito formula for the symmetric polynomials of the principle curvatures” to be obtained (1 p.3). From this, researchers were able to determine that, “while the vector of all the symmetric polynomials of the principle curvatures is a diffusion, the same was not true for any proper subset of the vector” (11 p.3). These models can be further extended by looking at the effervescent behavior of the Lipschitz-Killing curvatures of arbitrarily developing manifolds under isotropic and volume preserving Brownian flows on Rn (1). The relevant equation for the Lipschitz-Killing curvatures {?k (M)}dim(M) k=0, which is additionally regarded as curvature measures, can also be interpreted as augmentations of intrinsic volumes, which can also be called generalized volumes (1). The details of the Lipschitz-Killing curvatures can be symbolic representations of “the average of the symmetric polynomials of the principle curvatures over the manifold” (1 p.3), which, unlike the details provided by the indiscriminate filtration obtained by studying Lyapunov exponents that only gives limited information, Lipschitz-Killing curvatures describe the global geometry of randomly evolving manifolds (1). Additionally, we will examine the specifications of the most common axis. The most straight-forward type of coordinate system is a Cartesian system, which is a coordinate system that consists of three mutually perpendicular axes, the x-, y-, and z-axes (5). A quantity which possesses both magnitude and direction is termed a vector. By contrast, a quantity which possesses only magnitude is termed a scalar. Mass and time are scalar quantities. However, in general, displacement is a vector. Covariance functions that are not smooth on the diagonal, like “covariance associated with Sobolev norms of order between d/2 and (d + 2)/2, d being the dimension of the space” (8 p.651) have the capability to produce Wiener solutions which describe arbitrary evolutions of different varieties, such as “turbulent evolutions where (a) is not satisfied, which means that two points thrown initially at the same place separate, even when there is no pure diffusion, such that and coalescing evolutions where (b) does not hold” (8 p.651). Adopting a more general approach based on consistent systems of n-point as determining the law of the motion of n indivisible points thrown into the fluid, standard and combining evolutions are represented by flows of maps, and chaotic evolutions by flows of possibility cores that describe how a point mass composed of a continuum of inseparable points in x at time s is spread at time t (8). However, in this instance, the motion of an indissoluble point has not been fully determined by the flow (8). When dealing with turbulent evolutions, it is necessary to distinguish the transitional ones in which two points placed in the fluid at congruent points separate but can meet later, even in cases where (a) and (b) are both not satisfied (8). These flows can usually be paired with a coalescent flow (8). To further analyze the data, we created histograms over numerous trials, which estimated the intrinsic dimension the dataset oscillates between and which dataset deviates from the “informal” intrinsic dimension estimated (11). Finally, we applied the Euclidean method to a real data set, and, consequently chose the set of information from a publicly available database containing X subjects with different viewing conditions for each subject (7). The images were taken against a fixed background which we did not bother to segment out (7). We think this is justified since any fixed structures throughout the images would not change the intrinsic dimension or the intrinsic entropy of the dataset (7). We randomly selected 4 images from this data base and sub-sampled each configuration down to a pixels image and we then normalized the pixel values to create constant values that we could use to gauge the results (7). Objectives The main objectives of the research into the Geometric model for motion of curves specified by acceleration is to achieve the following: 1. The primary curvature properties of evolution of curves will be reviewed. 2. Derive time – evolution equations for the Frenet frame. This paper aims to demonstrate that the Frenet apparatus of an evolute curve can be formed according to apparatus of an involute curve and prove that there are presently no inclined evolutes in Minkowski space-time (11). 3. Derive time – evolution equations to satisfy the intrinsic quantities of curves. 4. Using the solution of the evolution equation for the curvature and torsion and 5. The Fundamental theorem for space curves to show the evolving space curve. Through a qualitative examination of relevant literature, this analysis aims to meet the primary objectives while providing: A. A working knowledge of Frenet, Gaussian, Brownian, and Hurstian formulations of curvature analysis. B. An understanding of variation principles in general and in mechanics in particular. C. Knowledge of a suite of exactly soluble dynamical systems, like the harmonic oscillator, the 2-body central force system, and other relative systems. D. A solid understanding of conservation laws, their utility, and their roots in symmetries of variation principles. E. Proficiency in mapping mechanical systems to mathematical representations and analyzing the resulting mathematical models. F. Aptitude with certain features of analytic geometry, vector analysis, and differential equations. G. A comprehensive understanding of scientific and mathematical written and oral description. Design The proposed research will rely on current literature to compile statistical models, aggregate data, such as formulas and equations, and working models for analysis. This project will look at solution flows of stochastic differential equations on non-compact manifolds and the properties of the solutions in terms of the geometry and topology of the underlying manifold itself. Some results on "strong p-completeness" are expected, given conditions on the derivative flow, and will be allowed suitable conditions on the coefficients of the stochastic differential equations. The most prominent theoretical frameworks will be used for computational analysis, including Frenet. Brownian, Bismut-Witten Laplacian, Gaussian, Hurst, and Minkowski. These formulas and equations will be studied to determine their relevancy when examined using contemporary datasets and comparison analyses will be conducted to determine which theoretical prospects satisfy the research question. For real time application, proposed algorithmic models will be applied to manifolds of known structure as well as real data sets containing various images. All the simulations used will be in regards to intrinsic dimension estimation and we will compare our algorithms to similarly indexes containing the real intrinsic dimension to our estimated ones and look at the residual errors as a function of subspace dimension. In this manner, we will validate the algorithms on standard synthetic manifolds in the literature, like linear planes in several dimensions, the 2-dimensional Swiss roll, and S-shaped surface. We will also apply our method to a high dimensional synthetic image data set, such as the same images generated by varying three different parameters: vertical and horizontal pose, and lighting direction. We will also use databases created for these functions, like ISOMAP, C-ISOMAP, or Hessian eigenmap and embed each image in the 3-dimensional Euclidean space using the common lexicographic order, applying the algorithms multiple times over each dataset to ensure accuracy. Estimated and real values will be plotted and the Euclidean model will be applied to the real dataset of unknown manifold structure and intrinsic aspect containing the images of varying dimensions. Research Plan We will design our research project according to the "research plan" and "time table" currently available. We will work according to the following: I. During the first two months of the work we will reviewing and collecting the literatures using on line mathematical data base, mathematical reviews, inter-library services and personal contacts with the working in the field. II. We will investigate deeply and sort out the useful results of our research. This may be take approximately two months. III. We will discuss known and new results which will give the utilities of work. It may take approximately four months. IV. Finally, we will give an example of application. That may require one month. V. Writing of the material and typing may take one month of the project. Anticipated Outcome This research is motivated by the question: How is the motion of manifolds specified by acceleration described? and seeks to determine the geometric model for motion of curves specified by acceleration. This is accomplished qualitative research and thorough examinations, conducted over time. The expected results of this study is to provide confirmation of the determinations regarding tried theoretical models as applied to current geometrical representations in 3D computer simulations. Additionally, examining existing theoretical principles using aggregate data that represents more complete geometric patterns will enable the compilation of more precise information regarding the geometric model for the motion of curves. In-depth analyses regarding the principles of acceleration as it relates to the motion of curves and manifolds can provide valuable information to many professions that encounter the variables created from the motion of curves specifically relative to acceleration. There are limitations to the results quantified from this study in that this formulation nominally requires computing diffusion in two spatial dimensions higher than the original problem itself, meaning that a moving curve is inherently a one spatial dimension problem. However, in cases where computational efficiency becomes critical, like when higher dimensional surfaces are encountered, the diffusion calculations can be restricted to a neighborhood of the surface or the vicinity of the curve. This ensures that the effective dimension of the computational problem approximates that of the intrinsic problem, although development of specific efficient algorithms remains to be carried out. Certain issues are unique to this new setting, and require further exploration rather than mere generalization of the original properties of relative to the Geometric Model for Motion of Curves Specified by Acceleration. An additional challenge presents itself in determining how to impose boundary conditions on the moving curve C at boundaries or edges in the underlying surface, S. We must also take particular interest in conditions that will allow all geodesics to become immobile solutions. A suitable extension to surfaces that cannot be isometrically embedded in without self-intersection also remains undeveloped. However, in such cases it may be possible to apply the exact same algorithm within a four-dimensional Euclidean space or greater, in which the surface is embedded without intersection. Another unresolved problem arises due the characterization of the motions laws concerning linear diffusion when it is replaced by convolution with a general smoothing kernel (9). However, if it were possible to extend the ergodic properties of a solution this would be a major step in the direction of improving the results obtained (8). An additional limitation may arise in the accuracy of the estimations made regarding the data analysis. In relation to the estimates, one of the directions of future research in this field would be to try to prove ergodicity for the flow (8). With appropriate conditions on the vector fields, the flow will traverse almost all the points in the space and can be believed to exhibit ergodicity (8). If we want to minimize the change in acceleration of our motion we apply the variational subdivision scheme in the non-uniform parameter setting (7). The parameterization we use is a centripetal parameterization derived with help of the image points presents a non-uniform rigid body motion of the interpolating input positions (10). Another aspect for consideration is that a Euclidean body in motion can display a rigid body computed with interpolatory variational subdivision in the uniform and non-uniform parameter setting (10). The acceleration of these two motions has been estimated by the arithmetic mean of the squared acceleration of the feature curves used in the computation (10). We can compare the acceleration of the two motions and see that the acceleration varies less in the non-uniform parameter setting and, using the approximating variational subdivision algorithm for curves, we can create motions that interpolate or approximate the established input positions according to the choice of the parameters ?i (7). Finally, this research analysis seeks to emphasize that the curvature motion of curves on surfaces can be derived via standard three-dimensional diffusion, which will allow determinations to be made concerning how the motion of manifolds specified by acceleration can be described. Alternatively, we can interpret three-dimensional diffusion coupled with the closest point extension procedure as a simple way to calculate the Geometric Model for Motion of Curves Specified by Acceleration and carry out the intrinsic surface diffusion process. However, we also note that only a few of the many computational algorithms have been explored and there are various other models that can be used in such determinations. This provides ample room for further exploration into this topic. References 1) Vadlamani S. On the diffusion of shape [PhD Thesis]. The Technion – Israel Institute of Technology Heshvan; 2007 [cited 2011 Oct 12]. Available from: http://math.tifrbng.res.in/~sreekar/Site/Personal_files/thesis_1.pdf 2) Howard P. Partial differential equations in MATLAB 7.0. 2005 [cited 2011 Oct 12]. Available from: http://www.math.tamu.edu/~phoward/m401/pdemat.pdf 3) Cipolla R. The visual motion of curves and surfaces. Phil. Trans. R. Soc. Lond. A [Internet]. 1996 [cited 2011 Oct 12];1-16. Available from: http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.7570 4) Introduction:The Particle-Curves, velocity, acceleration. The First law. Physics 3550. 2011 [cited 2011 Oct 12]. Available from: http://www.physics.usu.edu/torre/3550_Fall_2011/Lectures/01.pdf 5) Fitzpatrick R. Classical mechanics: An introductory course. The University of Texas at Austin [Internet]. [cited 2011 Oct 12]. Available from: http://farside.ph.utexas.edu/teaching/301/301.pdf 6) Tazawa Y. Theory of Curves and Surfaces: An Introduction to Classical Differential Geometry by Mathematica (in Japanese ) [Internet]. Pearson Education; 1999. Excerpt. [cited 2011 Oct 12]. Available from: http://math.kn.dendai.ac.jp/tazawa/HomePageFiles/courseware.pdf 7) Fowler M. Brownian Motion [Internet]. 2008 [cited 2011 Oct 12]. Available from: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/BrownianMotion.htm 8) Le Jan Y. New developments in stochastic dynamics. Paper presented at: Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2000. [Internet]. [cited 2011 Oct 12]. Available from: http://www.icm2006.org/proceedings/Vol_III/contents/ICM_Vol_3_34.pdf ref 34 9) Gilmore R. Lie Groups, Physics, and Geometry [Internet]. 2008 [updated 2008, Mar 11; cited 2011 Oct 12]. Cambridge University Press. Available from: http://einstein.drexel.edu/~bob/LieGroups.html 10) Rose O. Estimation of the Hurst Parameter of Long Range Dependent Time Series [Internet]. University of Wurzburg, Institute of Computer Science Research Report Series. 1996 [cited 2011 Oct 12]. Available from: http://www.eecis.udel.edu/~mills/fractal/tr137.pdf 11) Turgut M., Yilmaz S. On the Frenet Frame and a Characterization of Space-like Involute-Evolute Curve Couple in Minkowski Space-time. International Mathematical Forum [Internet]. 2008 [cited 2011 Oct 12]; 3(16):793 – 801. Available from: http://www.m-hikari.com/imf-password2008/13-16-2008/turgutIMF13-16-2008-2.pdf 12) http://www.xuemei.org/Xue-Mei.Li-thesis.pdf ref 35 Budget PROJECT TITLE Ruled Surfaces in Analogy to Bertrand Curves and their Applications DURATION MONTHS Item Category # Compensation FIRST YEAR SECOND YEAR TOTAL Months Budget Months Budget MANPOWER PI 1 12 14400 14400 Co-investigators 1 12 12000 12000 Research Assistants Students Technicians Consultants Secretary/Clerical Others Summer Compensation Total Salaries (Including Summer compensation) 26400 EQUIP & MATERIAL  Major Equipment (< = 100.000)   Equipment (> 100,000)  Materials & Supplies 5000 ITEM TOTAL 5000 TRAVEL  Training 10000 Field Trips 10000 ITEM TOTAL 20000 OTHERS  Patent registration  Publications 2000  Workshop Other Expenses 3000 ITEM TOTAL 5000 GRAND TOTAL 56400 Read More