Implicit Solutions of PDE's

Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a single unknown $\phi$ are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables, whose coefficients are functions of $\phi$ is discussed, and it is shown that if the discriminant of the quadratic vanishes, then an implicit solution of the so-called Universal Field Equation is obtained. The relation to the general solution is discussed.Comment: 11 pages LaTeX2

Universal Field Equations with Reparametrisation Invariance

New reparametrisation invariant field equations are constructed which describe $d$-brane models in a space of $d+1$ dimensions. These equations, like the recently discovered scalar field equations in $d+1$ dimensions, are universal, in the sense that they can be derived from an infinity of inequivalent Lagrangians, but are nonetheless Lorentz (Euclidean) invariant. Moreover, they admit a hierarchical structure, in which they can be derived by a sequence of iterations from an arbitrary reparametrisation covariant Lagrangian, homogeneous of weight one. None of the equations of motion which appear in the hierarchy of iterations have derivatives of the fields higher than the second. The new sequence of Universal equations is related to the previous one by an inverse function transformation. The particular case of $d=2$, giving a new reparametrisation invariant string equation in 3 dimensions is solved.Comment: 9page

Integrable Top Equations associated with Projective Geometry over Z_2

We give a series of integrable top equations associated with the projective geometry over Z_2 as a (2^n-1)-dimensional generalisation of the 3D Euler top equations. The general solution of the (2^n-1)D top is shown to be given by an integration over a Riemann surface with genus (2^{n-1}-1)^2.Comment: 8 pages, Late

A Model for Classical Space-time Co-ordinates

Field equations with general covariance are interpreted as equations for a target space describing physical space time co-ordinates, in terms of an underlying base space with conformal invariance. These equations admit an infinite number of inequivalent Lagrangian descriptions. A model for reparametrisation invariant membranes is obtained by reversing the roles of base and target space variables in these considerations.Comment: 9 pages, Latex. This was the basis of a talk given at the Argonne National Laboratory 1996 Summer Institute : Topics on Non-Abelian Duality June 27-July 1

The Reversed q-Exponential Functional Relation

After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.Comment: 6 pages, LaTeX, no figure

The Kauffman Index: Startup Activity National Trends - 2016

This report provides a broad index measure of business startup activity in the United States. It is an equally weighted index of three normalized measures of startup activity:The Rate of New Entrepreneurs in the economy, calculated as the percentage of adults becoming entrepreneurs in a given month.The Opportunity Share of New Entrepreneurs, calculated as the percentage of new entrepreneurs driven primarily by "opportunity" vs. "necessity."The Startup Density of a region, measured as the number of new employer businesses normalized by total business population

The Kauffman Index: Startup Activity - Metropolitan Area and City Trends 2015

How can I actually measure the entrepreneurial activity in my region? This is a question we at the Kauffman Foundation often hear from economic and policy leaders. As cities around the globe rally to foster entrepreneurship, the challenge of how to consistently measure and benchmark progress remains largely unanswered. While anecdotal evidence abounds, most ecosystems struggle to answer straightforward, yet often elusive, questions: How many new startups does our city or state have? How much are our ventures growing? How many of our businesses are surviving? To begin to answer these questions and address this challenge, we introduce the new Kauffman Index of Entrepreneurship, the first and largest index tracking entrepreneurship across city, state, and national levels for the United States. In this release, we introduce the Kauffman Index: Startup Activity—the first of various research installments under the umbrella of the new Kauffman Index of Entrepreneurship.For the past ten years, the original Kauffman Index— authored by Robert W. Fairlie—has been an early indicator for entrepreneurship in the United States, used by entrepreneurs and policymakers, from the federal to state and local levels. The Kauffman Index also has been one of the most requested and far-reaching entrepreneurship indicators in the United States and, arguably, the world. In the policy world, the Index has been referenced in multiple testimonies to the U.S. Senate and House of Representatives, by U.S. Embassies and Consulates across various countries—including nations like Spain, Ukraine, and United Kingdom—by multiple federal agencies, by state governments and governors from fifteen states— from Arizona to New York—and by the White House's office of the President of the United States. On the academic side, more than 200 research papers quote the Kauffman Index. In media circles, the Kauffman Index has been highlighted in more than 100 media channels, including most major publications like The New York Times, The Wall Street Journal, TIME, CNN, the Financial Times, and Harvard Business Review. Originally, the Kauffman Index tracked one of the earliest measures of business creation: When and how many people first start working for themselves, becoming entrepreneurs. Now, we are expanding it to include other dimensions of entrepreneurship. The new and expanded Kauffman Index of Entrepreneurship 2015 remains focused primarily on entrepreneurial outcomes, as opposed to inputs. That means we are more concerned with actual results of entrepreneurial activity—things like new companies and growth rates.The Kauffman Index: Startup Activity algorithm presented in this report takes into account three variables:* Rate of New Entrepreneurs* Opportunity Share of New Entrepreneurs* Startup Densit

Gauge theories and non-commutative geometry

It is shown that a $d$-dimensional classical SU(N) Yang-Mills theory can be formulated in a $d+2$-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. In this paper we present an explicit proof for the case of the torus and the sphere.Comment: 12 page

Wigner Trajectory Characteristics in Phase Space and Field Theory

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple---indeed, classical---for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.Comment: 9 pages, LaTex2