Prof. Michael Berman

Fractional Calculus

Driven-damped oscillator and its general closed form exact solution: New questions in fundamental physics and in other fields, which cannot be formulated adequately using traditional integral and differential calculus emerged recently. Fractional calculus was shown to describe phenomena where conventional approaches have been unsatisfactory. The driven damped fractional oscillator entails a rich set of important features, including loss of energy to the environment and resonances. In this paper, this oscillator with Caputo fractional derivatives is solved analytically in closed form. The exact solution is expressed in terms of generalized Mittag-Leffler functions. The standard driven damped Harmonic Oscillator is recovered as a special case of non-fractional derivatives. In contradistinction to the standard oscillator, the solution of the fractional oscillator is shown to decay algebraically and to possess a finite number of zeros. Several decay patterns are uncovered and are a direct consequence of the asymptotic properties of the generalized Mittag-Leffler functions. Other interesting properties of the fractional oscillator like the momentum–position phase-plane diagrams and the time dependence of the energy terms are discussed as well.

Exceptional points in the Riesz-Feller Hamiltonian with an impenetrable rectangular potential: The number of bound states in a standard rectangular potential well depends on the potential depth and width. In an impenetrable one-dimensional rectangular potential well, there are infinite bound states. In this work we study a non-Hermitian Riesz-Feller kinetic energy; i.e., the second-order derivative of the standard kinetic energy operator is replaced by a fractional, αth-order derivative. We show that for α < 2 a particle in an impenetrable one-dimensional rectangular potential well contains a finite number of bound states and an infinite number of metastable decaying states. The transitions from bound states to metastable decaying states occur at α values that correspond to exceptional points, for which two bound states coalesce. Our findings indicate that one can describe a transition of highly excited bound states to metastable decaying states, for example due to the interactions of atoms and molecules with the environment, by using the Riesz-Feller kinetic energy operator rather than the standard one.

Ultrasound of the Breast

Breast cancer is one of the leading causes of death from cancer. Early detection is widely believed to reduce breast cancer mortality by allowing intervention at an earlier stage of cancer progression. Screening [X-ray] mammography has secured a place as the gold standard routine health maintenance procedure for women – this is a mature technology that provides high-quality images in the majority of patients. However, conventional mammography does not detect all breast cancers, including some that are palpable, and as many as three-quarters of all breast lesions biopsied because of a suspicious finding on a mammogram turn out to be benign. The purpose of this research is to find alternative solutions to early detection of breast cancer.

Ultrasound Beamforming and Ultrasound Tomography

The conventional ultrasound approach is driven by the need for real-time data acquisition and display. Therefore, some of the complex physics associated with propagation of sound waves is traded off. One of the tradeoffs corresponds to the usage of straight-ray theory, a basic approximation of the true physics of acoustic wave propagation, which is only valid for purely homogenous media. A second important tradeoff is the assumption of a 2 dimensional geometry in which only the directly backscattered reflections are collected. The purpose of this research is to implement beamforming and tomography approaches that “undo” the trade-offs of conventional ultrasound, leading to a marked increase in the signal-to-noise ratio, while reducing artifacts and yielding higher quality images for greater clinical “sensitivity”. Furthermore, the signals that propagate through the anatomy, and which are never reflected back, contain additional information.

Technology and Academic Contributions

Michael has developed technology, issued patents, and published scientific papers and technology reports in the following subjects:

- Fractional Calculus
- Ultrasound Tomography
- Segmentation of Ultrasound Images
- Fetal Weight Determination with Ultrasound
- The Heart in Ultrasound Imaging
- Guidance of Minimally Invasive Surgical Procedures using Real-Time 3D Ultrasound
- Thermo Field Dynamics
- Fast Reactions in Liquids
- Time-dependent Solution of the Liouville von Neumann Equation
- Self-Consistent Quantal Equations of Motion within the Lie Algebraic Setting
- Electron Molecule Scattering
- Theory of the Double Resonance Raman Amplifier
- Excited States of Rare-Gas Molecules
- Statistical Mechanics of Adsorption of Large Molecules on Charged Electrodes

M. Berman and U. Kaldor, "Fast Calculation of Excited State Potentials for Rare Gas Diatomic Molecules: Ne2 and Ar2'', Chem. Phys. 43, 375-383 (1979)

Y Rabin, M. Berman and A Ben-Reuven, "Theory of the double resonance Raman amplifier", J. Phys. B: At. Mol. Phys. 13 2127-2136, (1980)

M. Berman and U. Kaldor, "Electron-Molecule Scattering with Polarization Using the Schwinger Variational Principle", Chem. Phys. Letters 79, 489-493 (1981)

M. Berman, U. Kaldor, J. Shmulovich and S. Yatsiv, "Rydberg States and the Observed Spectrum of ArH", Chem. Phys. 63, 165-173 (1981)

M Berman and U Kaldor, "The Schwinger variational method in electron-atom and electron-molecule scattering theory with polarization", J. Phys. B: At. Mol. Phys. 14 3993-4005, (1981)

M. Berman, O. Walter, and L. S. Cederbaum, "Electron-Molecule Scattering in the Optical-Potential Approach: Surpassing Second Order", Phys. Rev. Lett. 50, 1979–1982 (1983)

Michael Berman, Hernán Estrada, L. S. Cederbaum, and W. Domcke, "Nuclear dynamics in resonant electron-molecule scattering beyond the local approximation: The 2.3-eV shape resonance in N2", Phys. Rev. A 28, 1363–1381 (1983)

H. Estrada, M. Berman, L. S. Cederbaum and W. Domcke, “Theoretical study of electron transmission through N2”, Chem. Phys. Lett. 97, 352 (1983)

M Berman, L S Cederbaum and W Domcke, "Analysis of the ambiguities in the definition of the local complex potential in resonant electron-molecule scattering”, J. Phys. B: At. Mol. Phys. 16 875-890, (1983)

Michael Berman and Wolfgang Domcke, " Projection-operator calculations for shape resonances: A new method based on the many-body optical-potential approach", Phys. Rev. A 29, 2485–2496 (1984)

Michael Berman and Wolfgang Domcke, " Projection-operator calculations for shape resonances: A new method based on the many-body optical-potential approach", Phys. Rev. A 29, 2485–2496 (1984)

M Berman and W Domcke, "Direct calculation of complex resonance poles in electron-molecule scattering using separable T-matrix expansions", J. Phys. B: At. Mol. Phys. 17 L453-L458, (1984)

Claus Mündel, Michael Berman, and Wolfgang Domcke, "Nuclear dynamics in resonant electron-molecule scattering beyond the local approximation: Vibrational excitation and dissociative attachment in H2 and D2", Phys. Rev. A 32, 181–193 (1985)

W. Domcke, M. Berman, H. Estrada, C. Muendel, L. S. Cederbaum, “Aspects of nuclear dynamics in short-lived negative ion states”, J. Phys. Chem., 1984, 88 (21), pp 4862–4867

Michael Berman, Claus Mündel, and Wolfgang Domcke, "Projection-operator calculations for molecular shape resonances: The 2 Sigma +u resonance in electron-hydrogen scattering", Phys. Rev. A 31, 641–651 (1985)

Robert B. Gerber, Ronnie Kosloff and Michael Berman, "Time-Dependent Wavepacket Calculations of Molecular Scattering from Surfaces", Comp. Phys. Rep., 5,59-114 (1986)

W. Domcke, M. Berman, C. Mündel, and H.-D. Meyer, “Direct calculation of complex resonance poles using separable expansions of the potential: Application to the 2Σu+ shape resonance in electron-H2 scattering”, Phys. Rev. A 33, 222–232 (1986)

Michael Berman, "Thermal quasiparticles with phase-space distribution ", Phys. Rev. A 40, 2057–2063 (1989)

D. Charutz, M. Berman and R.D. Levine, "An Exponential Gap Relation for Vibrational Energy Transfer in a Dissipative Medium", Chem. Phys. Lett. 164, 495 (1989)

Michael Berman, "Wigner phase-space representation of thermal excitations”, Phys. Rev. A 42, 1863–1868 (1990)

Michael Berman and Ronnie Kosloff, "Time Dependent Solution of the Liouville Von Neumann Equation: Non-Dissipative Evolution", Comp. Phys. Comm., 63, 1-20 (1991)

Michael Berman, Ronnie Kosloff and Hillel Tal-Ezer, "Solution of the time dependent Liouville von Neumann Equation Dissipative evolution", J. Phys. A, 25,1283-1307 (1992)

Michael Berman and Lorenz S. Cederbaum, “Fractional driven-damped oscillator and its general closed form exact solution”, Physica A 505, 744–762 (2018)

Michael Berman and Nimrod Moiseyev, “Exceptional points in the Riesz-Feller Hamiltonian with an impenetrable rectangular potential”, Physical Review A 98, 042110 (2018)

Papers and Abstracts - Proceedings of Conferences

M. Zipparo, C. Oakley, R. Denny, S. Azim, V. Balannik, Z. Soferman, M. Berman, R. Nechushtai and D. Kopelman, “3-D Laparoscopic Imaging”, 2008 IEEE International Ultrasonics Symposium Proceedings, 40-44.

Patents

Berman Michael, Gessert James, Moore Wayne, Nechushtai Rachel, Rom Hillel, and Soferman Ziv, “3-dimensional ultrasonic imaging”, US Patent 6,315,724, November 13, 2001

Ziv Soferman and Michael Berman, “Automatic fetal weight determination”, US Patent 6,375,616, April 23, 2002

Ziv Soferman and Michael Berman, “Facial imaging in utero”, US Patent 6,434,260, August 13, 2002 
Ziv Soferman and Michael Berman, “Determination of fetal weight in utero”, US Patent 6,575,907, June 10, 2003

Michael Berman, “System and Method for Ultrasound Examination of the Breast”, WIPO Patent WO/2012/077111A1, June 14, 2012

Michael Berman, “System and Method for Ultrasound Examination of the Breast”, Patent Certificate of Chinese Invention, Patent No. 201180066669.5, May 5, 2015

Technical Reports

Michael Berman, Vadim Balannik and Bill Glenn, "Guidance of Minimally Invasive Surgical Procedures using Real-Time 3D Ultrasound", Technical Report,BIRD Project Ref. No.: 1109 (January 2008)

Michael Berman, Vadim Balannik and Bill Glen, "Guidance of Minimally Invasive Surgical Procedures using Real-Time 3D Ultrasound", Technical report, BIRD Project Ref. No.: 1109, (December 2008)

Michael Berman and Lori Stayton, "Guidance of Minimally Invasive Surgical Procedures using Real-Time 3D Ultrasound", Technical report, BIRD Project Ref. No.: 1109, (April 2009)

Michael Berman and Said Azim, "Guidance of Minimally Invasive Surgical Procedures using Real-Time 3D Ultrasound", Technical report, BIRD Project Ref. No.: 1109, (December 2009)

Ilan Sinai, Meir Friedman, Ziv Soferman and Michael Berman, "Ultrasound Tomography", White Paper, (April 2010)

Michael Berman and Ilan Sinai, "Tomographic Ultrasound of the Breast", Technical report, OCS Project No.: 40930, (May 2010)

Michael Berman, "Tomographic Ultrasound of the Breast", Technical report, OCS Project No.: 40930, (October 2011)

Michael Berman "Early detection of breast cancer using ultrasound tomography", Healthymagination Challenge, a General Electric Web Publication, November 2, (2011)

Michael Berman, Astrid Duerenberger, Lea Even, Irit Levy-Feldman, Lydia Linortner, Sara Meilijson, Jean-Luc Patry, Michael Weingarten, “Learning to Be, Lifelong Learning in the 21st Century” LLAF Tempus IV (2017)

Michael Berman and Miki Harris, “Lifelong Learning in Applied Fields (LLAF) Final Report on implementation of the project, Summary report for publication”, Tempus IV, Sixth Call for proposals EACEA no. 35/2012, 543894-TEMPUS-1-2013-1-IL, (2017)