Commutative law for products of infinitely large isotropic random matrices

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N \rightarrow \infty$. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power A^n of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as of A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \rightarrow 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N \rightarrow \infty$: the eigenvalue density of an isotropic random matrix has a power law singularity at the origin $\sim |\lambda|^{-s}$ with a power $s \in (0,2)$ when and only when the density of its singular values has a power law singularity $\sim \lambda^{-\sigma}$ with a power $\sigma = s/(4-s)$. These results are obtained analytically in the limit $N \rightarrow \infty$. We supplement these results with numerical simulations for large but finite N and discuss finite size effects for the most common ensembles of isotropic random matrices.Comment: 15 pages, 4 figure

New spectral relations between products and powers of isotropic random matrices

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed non-hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide an evidence that the result holds also for isotropic orthogonal ensembles (IOE).Comment: 8 pages, 3 figures (in version 2 we added a figure and discussion on finite size effects for isotropic orthogonal ensemble

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical Physics and Information Theory," September 26-30, 2010, Krakow, Polan

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices

We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.Comment: 13 pages, 3 figure

Experimental and economic evaluation of different culture systems for mesenchymal stromal/stem cell expansion for clinical applications

The translation of cell therapies into clinical practice requires a scalable, efficient and cost-effective manufacturing process. This study presents an integrated experimental and cost analysis of different cell culture technologies for commercial manufacture of a novel umbilical cord-derived cell therapy, currently in early phase clinical trials for the treatment of acute graft-versus-host disease (aGvHD). The experimental analysis assessed the expansion and harvest potential of mesenchymal stromal cells (MSCs), derived from umbilical cord matrix (UCM-MSCs), in different scalable cell culture technologies: a multi-layer vessel (ML), a stirred tank bioreactor with microcarrriers (STR), a hollow fiber bioreactor (HF) and a packed-bed bioreactor (PB). The presentation will highlight differences in cell proliferation rate, expansion fold and harvesting efficiency across the technologies. The cells retained their functional properties post culture in all the technologies evaluated. The experimental results were incorporated into a bioprocess economics tool comprising a stochastic cost of goods (COG) and sizing model to evaluate the commercial economic feasibility and robustness of the technologies. The financial and risk rank orders predicted by the tool will be presented, as well as their sensitivity to the reimbursement scenario selected. The model determined industrially relevant scenarios for which no technology will yield a satisfactory gross margin, indicating that many studies are still needed to establish an optimized manufacturing process

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.Comment: 20 pages, v2 published version: typos corrected and references adde

Opa vs Oper: Neutralization of /?/ and unstressed /a/ contrast in a perception and production study

The present study examined differences in production and perception of the German vowels /a/ and /?/ in word-final, unstressed position. In the first experiment, 3 male and 3 female speakers produced minimal pairs embedded in meaningful sentences and varied in prosodic environment. In the second experiment, the minimal pairs were extracted from the context and presented to 44 listeners for a forced-choice identification task. Results showed a better-than-chance performance that was, however, mainly driven by one male speaker. Temporal and spectral measures confirmed that only this speaker produced an acoustic difference between /a/ and /?/

CRISPR-Cas9 Knockin Mice for Genome Editing and Cancer Modeling

CRISPR-Cas9 is a versatile genome editing technology for studying the functions of genetic elements. To broadly enable the application of Cas9 in vivo, we established a Cre-dependent Cas9 knockin mouse. We demonstrated in vivo as well as ex vivo genome editing using adeno-associated virus (AAV)-, lentivirus-, or particle-mediated delivery of guide RNA in neurons, immune cells, and endothelial cells. Using these mice, we simultaneously modeled the dynamics of KRAS, p53, and LKB1, the top three significantly mutated genes in lung adenocarcinoma. Delivery of a single AAV vector in the lung generated loss-of-function mutations in p53 and Lkb1, as well as homology-directed repair-mediated Kras[superscript G12D] mutations, leading to macroscopic tumors of adenocarcinoma pathology. Together, these results suggest that Cas9 mice empower a wide range of biological and disease modeling applications.National Science Foundation (U.S.). Graduate Research Fellowship (Grant 1122374)Damon Runyon Cancer Research Foundation (Fellowship DRG-2117-12)Massachusetts Institute of Technology. Simons Center for the Social Brain (Postdoctoral Fellowship)European Molecular Biology Organization (Fellowship)Foundation for Polish Science (Fellowship)American Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipNational Science Foundation (U.S.). Graduate Research FellowshipMassachusetts Institute of Technology (Presidential Graduate Fellowship)Human Frontier Science Program (Strasbourg, France) (Postdoctoral Fellowship)National Human Genome Research Institute (U.S.) (CEGS P50 HG006193)Howard Hughes Medical InstituteKlarman Cell ObservatoryNational Cancer Institute (U.S.) (Center of Cancer Nanotechnology Excellence Grant U54CA151884)National Institutes of Health (U.S.) (Controlled Release Grant EB000244)National Heart, Lung, and Blood Institute (Program of Excellence in Nanotechnology (PEN) Award Contract HHSN268201000045C)Massachusetts Institute of Technology (Poitras Gift 1631119)Stanley CenterSimons Foundation (6927482)Nancy Lurie Marks Family Foundation (6928117)United States. Public Health Service (National Institutes of Health (U.S.) R01-CA133404)David H. Koch Institute for Integrative Cancer Research at MIT (Marie D. and Pierre Casimir-Lambert Fund)MIT Skoltech InitiativeNational Cancer Institute (U.S.) (Koch Institute Support (Core) Grant P30-CA14051)National Institute of Mental Health (U.S.) (Director’s Pioneer Award DP1-MH100706)National Institute of Neurological Disorders and Stroke (U.S.) (Transformative R01 Grant R01-NS 07312401)National Science Foundation (U.S.) (Waterman Award)W. M. Keck FoundationKinship Foundation. Searle Scholars ProgramKlingenstein FoundationVallee FoundationMerkin Foundatio

Long-Term Continuous Corticosterone Treatment Decreases VEGF Receptor-2 Expression in Frontal Cortex

Objective: Stress and increased glucocorticoid levels are associated with many neuropsychiatric disorders including schizophrenia and depression. Recently, the role of vascular endothelial factor receptor-2 (VEGFR2/Flk1) signaling has been implicated in stress-mediated neuroplasticity. However, the mechanism of regulation of VEGF/Flk1 signaling under longterm continuous glucocorticoid exposure has not been elucidated. Material and Methods: We examined the possible effects of long-term continuous glucocorticoid exposure on VEGF/Flk1 signaling in cultured cortical neurons in vitro, mouse frontal cortex in vivo, and in post mortem human prefrontal cortex of both control and schizophrenia subjects. Results: We found that long-term continuous exposure to corticosterone (CORT, a natural glucocorticoid) reduced Flk1 protein levels both in vitro and in vivo. CORT treatment resulted in alterations in signaling molecules downstream to Flk1 such as PTEN, Akt and mTOR. We demonstrated that CORT-induced changes in Flk1 levels are mediated through glucocorticoid receptor (GR) and calcium. A significant reduction in Flk1-GR interaction was observed following CORT exposure. Interestingly, VEGF levels were increased in cortex, but decreased in serum following CORT treatment. Moreover, significant reductions in Flk1 and GR protein levels were found in postmortem prefrontal cortex samples from schizophrenia subjects. Conclusions: The alterations in VEGF/Flk1 signaling following long-term continuous CORT exposure represents a molecula