THE INVERSE PROBLEM FOR BOOLEAN EQUATIONS

Let us denote by CK the standard Cohen algebra of 7r-weight K, i.e. the complete Boolean algebra adjoining”‘ Cohen reals, where K is an infinite cardinal or 0. More generally, we call a Boolean algebra A a Cohen algebra if (for technical convenience in Theorems 0.3 and 0.4 below) it satisfies the countable chain condition and forcing with A (more precisely with the partial ordering A\ (O) is equivalent to Cohen forcing, i.e. if every generic extension of the universe of set theory arising from forcing with A arises from forcing with some standard Cohen algebra. Since forcing with an arbitrary Boolean algebra is equivalent to forcing with its completion and forcing with a product of algebras is equivalent to forcing with one of the factors, an algebra is Cohen iff its completion is isomorphic to a product of at most countably many standard Cohen algebras; we will use this description as the definition of a Cohen algebra in the rest of the paper.

In this paper, we propose distributed algorithms that solve a system of Boolean equations over a network, where each node in the network possesses only one Boolean equation from the system. The Boolean equation assigned at any particular node is a private equation known to this node only, and the nodes aim to compute the exact set of solutions to the system without exchanging their local equations. We show that each private Boolean equation can be locally lifted to a linear algebraic equation under a basis of Boolean vectors, leading to a network linear equation that is distributedly solvable. A number of exact or approximate solutions to the induced linear equation are then computed at each node from different initial values. The solutions to the original Boolean equations are eventually computed locally via a Boolean vector search algorithm. We prove that given solvable Boolean equations, when the initial values of the nodes for the distributed linear equation solving step are i.i.d selected according to a uniform distribution in a high-dimensional cube, our algorithms return the exact solution set of the Boolean equations at each node with high probability.

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Typically, simultaneous Boolean equations are expressed by a set of relations Fi = fi, i = 1,2 ?, where the Fi and fi are switching functions of N binary variables. In many logical design problems, a special type of Boolean equation is often encountered; namely, F(A0, A1 ?, An?1, X0, X1 ?, Xm?1) = f(a1, a2 ?, ak, X0, X1 ?, Xm?1), in which the arguments Ai's and Xi's are binary variables and the a1's are implicit functions of the Ai's only. This paper presents a somewhat new approach to solving such special types of Boolean functional equations. The solution function (see reference 1) can be formulated and mapped on a Veitch chart, especially tailored to the present problem, and all possible sets of solutions of the implicit functions ai's which satisfy the original functional equation are obtained simultaneously. Other possible applications and extensions of this method are discussed also.

Babbitt's Canonical Form Revisited:Codes and Metaphors for Epistemology Joshua Banks Mailman (bio) Even among Milton Babbitt's composition titles (such as It takes Twelve to Tango, Whirled Series, and the Joy of More Sextets) his title Canonical Form is unusually mysterious, not for being abstruse or impenetrable, but rather for being so multiply penetrable; that is, for being so easily unwrapped to reveal several possible meanings. Until recently, the titles of Babbitt's compositions have not been subject to much scrutiny, usually being treated merely as casual diversions, prompting mere casual speculation. Alison Maggart's (2017; 2021) recent work, however, delves into aspects of Babbitt's biography, identity, and personality to develop an unprecedented hermeneutical account of his music, foregrounding Babbitt's predilection for wordplay in the form of puns, considering the possible power dynamics of humor, Babbitt's Jewish identity, and his nostalgia for baseball. The present essay pursues such a hermeneutical account, focusing on Babbitt's solo piano work Canonical Form (1983). The third of Joseph Dubiel's "Three Essays on Milton Babbitt" (1990; 1991; 1992) addresses the overall grandeur of Babbitt's Canonical Form, suggesting some possible meanings of the title, which [End Page 215] provide part of the foundation that my remarks build on. Yet since the time of Dubiel's writing, not only have Babbitt's titles begun to be scrutinized more closely (Maggart 2017; 2021), but also a number of analytical technologies and attitudes have arisen which enable, or underwrite, an even broader spectrum of interpretations than was perhaps possible, or tenable. For instance, Gilles Fauconnier and Mark Turner's (1998; 2002) conceptual blending theory, including the analytical technology of conceptual integration networks (CINs), is enlisted to model the multifaceted, multivalent nature of Babbitt's music and titles.1 It provides a framework for interpreting various analytical results, as I've demonstrated before. For instance, as I showed previously (Mailman 2020) and show below, Schenkerian analysis can be deployed to reveal tonality in Babbitt's music, such that it can be read as a conceptual blend of serial and tonal syntaxes. Later in the analyses below I also apply, to pitch registers, both oscillator modeling and David Lewin's (1995) technique of binary state operators, which he had applied specifically to rhythms and instrumentation in two other Babbitt works. Finally I present a technique of cumulative pattern matching (across domains of pitch class and pitch register) based on Boolean functions (Boolean algebra) which reveals a hidden meaning in Babbitt's title. Yet another arises from an important matrix pattern in linear algebra. Ultimately, I suggest, as one aspect of Babbitt's compositional poetics, a certain pragmatism regarding ontologies in music theoretical discourse. That is, this piano work and its title, taken together, endorse a certain pragmatism as instrumental to the progress of musical inquiry. Canonicity One of the possible meanings of the title "canonical form" that is at least worth considering is the work's engagement with canonical repertoire, not in terms of anything as overt as quoting, but rather through subtler means. In this work, and others from the 1980s, Babbitt's pitch configurations dance with the tonal languages of canonic repertoire, by eluding to brief triadic chord progressions and even longer voice-leading motions. Canonical Form is based on the same array as Whirled Series, Quartet No. 5, and Around the Horn (Mead 1994; Dubiel 1997), an array whose row features three consonant triads as consecutive segments. Furthermore these works are based on superarrays, which enable a surprisingly vast flexibility in aligning pitches to create various configurations (Mailman 2019). This is because these superarrays feature three or more [End Page 216] simultaneous arrays whose ordering relative to each other is undetermined. Thus Babbitt can choose to align pitches from different array lynes, not only to forge consonant triads but also to suggest cadential tonal gestures and, in some cases, longer chromatic tonal progressions. A passage near the beginning of Canonical Form exemplifies this strikingly. Shown in Example 1, a perfect authentic (I-V-I) cadence in F# minor launches m. 21, prepared by a melodic lead-in (A and G# from m. 20). In m. 22, six pitches, drawn...

In the last 25 years, mathematical and numerical modeling of cardiovascular problems has experienced a spectacular progress. Moving progressively from the stage of idealized models supporting traditional in vitro and in vivo approaches – basically with proofs of concept – Cardiovascular Mathematics has been transforming into a unique tool for patient-specific analysis. This has been made possible by the strong integration with imaging devices. Nowadays, the work pipeline starting from a medical image in the Digital Imaging and Communications in Medicine (DICOM) format to retrieve the patient-specific geometry of an artery or the left ventricle to be used as computational domain in a numerical simulation is well established. This has initiated a paradigm shift in the interactions between mathematicians, bioengineers, and medical doctors. The latter are progressively getting used to in silico analysis as an important component of their research [3]. The introduction of numerical procedures as a part of an established clinical routine and more in general of a consolidated support to the decision-making process of physicians still requires some steps both in terms of infrastructures (to bring computational tools to the operating room or the bedside) and methods. In particular, the quality of the numerical results needs to be assessed and certified. The reliability of simulations calls for an accurate quantification and possibly reduction of uncertainty. In this scenario and in view of terrific advancements of measuring techniques, an important research line – quite established in other fields – is data assimilation [4,6,9], that is, the integration of numerical simulations and measurements. We may say that numerical models provide a background knowledge (based on physical principles and constitutive laws, not patient-specific), whereas measures give a foreground (individual) information; accuracy of in silico procedures relies on the correct integration of these two levels of knowledge of the problem [9,10]. As a matter of fact, numerical models depend on parameters and boundary/initial conditions. Their knowledge is in practice usually incomplete, inaccurate, and noisy. In some cases, the sensitivity itself of the final results to these parameters is unclear and needs to be quantified. Such an uncertainty may be significantly reduced by the availability of patient-specific measures; on the other hand, the quality of measures can be strongly enhanced by comparison with mathematical models (after all, noise is in general not educated, it does not know PDEs . . . ) [4,5,7,8,9,11]. We think therefore that, in the future, a crucial step for the clinical application of Cardiovascular Mathematics will be the passage from numerical simulation to data assimilation. To be concrete, borrowing a motto by Michel Fortin supporting mesh adaptivity in finite element simulations, stating that a ‘grid should not be considered a data, but an unknown of the problem’, we think that many parameters currently tuned on the basis of nonpatient specific criteria (even in patient-specific geometries) should be considered part of the unknowns to be found by assimilation procedures.

P 211. Efficient method for computing the TMS-induced electric field in spherically symmetric head models, the triangle Construction

Modeling ore generation in a magmatic context

BackgroundThe aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.MethodsWhile other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.ResultsIt starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.ConclusionSolving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.

Researchers have investigated the dynamics of oscillation of a single-link assistive manipulator and carried out vibration analysis of the same due to its great application in real life. Thus, studying the dynamics of flexible assistive robots and developing the model is indeed an interesting research topic. It is possible that the exact solution to the equations associated with such a single-link manipulator does not always exist. We used a numerical method, viz., Galerkin’s method to solve the governing differential equations of the problem, and the corresponding frequency of oscillation was calculated. This is known as the forward problem. On the other hand, the inverse problem is to compute values of the unknowns when the frequency is known by experiment, but the values of the remaining parameters are not known. With this backdrop, the purpose of this chapter is to investigate forward and inverse problems related to single-link robotic manipulators using Euler’s beam theory to define the system dynamics. Firstly, the frequencies of the first four modes of vibration have been calculated. Furthermore, the frequency is assumed to be known (experimental outcome), and the inverse case is studied using Young’s modulus and density of the material of manufacturing of the manipulator as unknown parameters. Besides, the system of single-link manipulators has been studied in uncertain environments using fuzzy theory for forward and inverse problems.

International Journal of Computational Engineering ScienceVol. 04, No. 01, pp. 19-65 (2003) No AccessFULLY NONLINEAR MODELING AND ANALYSIS OF PRECISION MEMBRANESP. FRANK PAI and LEYLAND G. YOUNGP. FRANK PAIDepartment of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Search for more papers by this author and LEYLAND G. YOUNGDepartment of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Search for more papers by this author https://doi.org/10.1142/S1465876303000788Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractHigh precision membranes are used in many current space applications. This paper presents a fully nonlinear membrane theory with forward and inverse analyses of high precision membrane structures. The fully nonlinear membrane theory is derived from Jaumann strains and stresses, exact coordinate transformations, the concept of local relative displacements, and orthogonal virtual rotations. In this theory, energy and Newtonian formulations are fully correlated, and every structural term can be interpreted in terms of vectors. Fully nonlinear ordinary differential equations (ODEs) governing the large static deformations of known axisymmetric membranes under known axisymmetric loading (i.e., forward problems) are presented as first-order ODEs, and a method for obtaining numerically exact solutions using the multiple shooting procedure is shown. A method for obtaining the undeformed geometry of any axisymmetric membrane with a known inflated geometry and a known internal pressure (i.e., inverse problems) is also derived. Numerical results from forward analysis are verified using results in the literature, and results from inverse analysis are verified using known exact solutions and solutions from the forward analysis. Results show that the membrane theory and the proposed numerical methods for solving nonlinear forward and inverse membrane problems are accurate.Keywords:Nonlinear membrane theorynumerically exact analysis References M. Salama , M. Lou and H. Fang , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 2000 ) . Google Scholar M. S. Smith , R. S. Schallenkamp and L. G. Seely , AIAA International Balloon Technology Conference . Google Scholar D. Gorinevsky , T. Hyde and C. Cabuz , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 2001 ) . Google Scholar V. Firt , Statics, Formfinding and Dynamics of Air-Supported Membrane Structures ( Martinus Nijhoff Publishers , Boston , 1983 ) . Google Scholar A. L. Palisoc, Inflatable Reflector Development Program, Task 3 Report, L'Garde Technical Report, LTR-94-AP-008, 1994 . Google Scholar S. V. Damle et al. , AIAA International Balloon Technology Conference . Google ScholarP. F. Pai, A. N. Palazotto and J. M. Greer Jr., Comput. Struct. 66, 823 (1998), DOI: 10.1016/S0045-7949(98)00004-2. Crossref, Google Scholar H. Furuya and Y. Kawasaki , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference . Google Scholar G. Greschik , M. M. Mikulas and A. Palisoc , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 1998 ) . Google ScholarJ. Ruze, Proceedings of the IEEE 54, 633 (1966), DOI: 10.1109/PROC.1966.4784. Crossref, Google Scholar A. Palisoc et al. , Fifth SPIE Conference on Space Telescopes and Instruments . Google ScholarP. F. Pai and A. H. Nayfeh, Nonlinear Dynamics 6, 459 (1994). Crossref, Google ScholarP. F. Pai and A. N. Palazotto, J. Eng. Mech. 121, 568 (1995), DOI: 10.1061/(ASCE)0733-9399(1995)121:4(568). Crossref, Google ScholarP. F. Pai and A. H. Nayfeh, Nonlinear Dynamics 2, 445 (1991), DOI: 10.1007/BF00045438. Crossref, Google Scholar K. Washizu , Variational Methods in Elasticity & Plasticity , 3rd edn. ( Pergamon Press Inc , New York , 1982 ) . Google ScholarP. F. Pai and A. N. Palazotto, Int. J. Solids Struct. 32, 3047 (1995), DOI: 10.1016/0020-7683(94)00273-Y. Crossref, Google Scholar H. Golden , T. W. Strganac and R. A. Schapery , AIAA International Balloon Technology Conference ( 1997 ) . Google Scholar J. Stoer and R. Bulirsch , Introduction to Numerical Analysis ( Springer-Verlag , New York , 1980 ) . Crossref, Google Scholar FiguresReferencesRelatedDetailsCited By 4Analytical Investigation of Dynamics of Inflatable Parabolic Membrane ReflectorFushou Liu and Dongping Jin1 Jan 2015 | Journal of Spacecraft and Rockets, Vol. 52, No. 1A point collocation method for geometrically nonlinear membranesKyle F. Kolsti and Donald L. Kunz1 Jan 2013 | International Journal of Solids and Structures, Vol. 50, No. 2Numerical and experimental dynamic characteristics of thin-film membranesLeyland G. Young, Suresh Ramanathan, Jiazhu Hu and P. Frank Pai1 May 2005 | International Journal of Solids and Structures, Vol. 42, No. 9-10Modeling, Analysis and Testing of some Deployable/Inflatable StructuresPerngjin Pai and Leyland Young26 June 2012 Recommended Vol. 04, No. 01 Metrics History Received 26 August 2002 Accepted 4 November 2002 KeywordsNonlinear membrane theorynumerically exact analysisPDF download

Many materials (e.g., rubber or biologic tissues) are "nearly" incompressible and often assumed to be incompressible in their constitutive equations. This assumption hinders realistic analyses of wave motion including acoustoelasticity. In this study, this constraint is relaxed and the reflected waves from nearly incompressible, hyper-elastic materials are examined. Specifically, reflection coefficients are considered from the interface of water and uni-axially prestretched rubber. Both forward and inverse problems are experimentally and analytically studied with the incident wave perpendicular to the interface. In the forward problem, the wave reflection coefficient at the interface is evaluated with strain energy functions for nearly incompressible materials in order to compute applied strain. For the general inverse problem, mathematical relations are derived that identify both uni-axial strains and normalized material constants from reflected wave data. The validity of this method of analysis is demonstrated via an experiment with stretched rubber. Results demonstrate that applied strains and normalized material coefficients can be simultaneously determined from the reflected wave data alone if they are collected at several different (but unknown) levels of strain. This study therefore indicates that acoustoelasticity, with an appropriate constitutive formulation, can determine strain and material properties in hyper-elastic, nearly incompressible materials.

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several ‘classical’ inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of ‘fractional’ inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.

'Acoustic probes have shown promise to be quite effective in determining the solid content in solid-liquid suspensions. However, the presence of small amounts of gas in the waste slurries stored in tanks across the DOE complex prevents straightforward application for characterization of these slurries. The proposed research will develop an acoustic probe for monitoring particle size and volume fraction in slurries in the absence and the presence of gas bubbles. Theoretical Analysis Accomplished: Attenuation of sound waves depends on the size distribution of the solids and the volume fraction of solids. These can in principle be calculated from attenuation measured over a range of frequencies. However, small amounts of bubbles distort the measured attenuation. A typical result from theoretical analysis for the attenuation of solid- gas-liquid systems is given in Figure 1. The total attenuation of a sound wave v(o) equals the sum of contributions by a large number of ''bins'' of particle sizes. This notion yields the following equation for the (hitherto) unknown number density of solid particles as a function of particle radius N(a): j k(o,a)N(a)da = v(o), where the kernel k(o,a) is obtained from analysis. If N(a) is given, the above equation is used to calculate the attenuation v(o). This is referred to as solving the ''forward problem''. Solving for N(a) with v(o) given is the ''inverse problem''. A complication that one faces when trying to solve the inverse problem is that the stated problem is mathematically ill-posed, i.e., small fluctuations in v(o) cause large fluctuations in the result for the number density. Therefore the problem needs to be ''regularized'', i.e., the stated problem needs to be changed slightly such as to make it well-posed. This has been done by others for gas-liquid systems in the past. This approach is currently being applied in the present project to solid-liquid systems. As is shown in Figure 2, it successfully recovers the number density that has been used in the forward problem to generate attenuation data. Having this solution technique giving reliable results for the inverse problems of both gas-liquid and solid-liquid systems, the authors shall apply this method in the near future to solid-gas-liquid systems.'

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which influence the extent to which the discretization impacts on the accuracy of the reconstructed images. For example, the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their orientation with respect to optical absorptive heterogeneities, and the formulation of the inverse problem. As a result, our error analysis shows that the discretization of one problem cannot be considered independent of the other problem. While our analysis focuses specifically on the discretization error in DOT, the approach can be extended to quantify other error sources in DOT and other inverse parameter estimation problems.