Stephen Wolfram

Wolfram Research

The Wolfram Physics Project is a bold effort to find the fundamental theory of physics. It combines new ideas with the latest research in physics, mathematics and computation in the push to achieve this ultimate goal of science. Written with Stephen Wolfram's characteristic expository flair, this book provides a unique opportunity to learn about a...

Through his pioneering work in science, technology and language design, Stephen Wolfram has developed his own signature way of thinking about an impressive range of subjects. In this lively book of essays, Wolfram takes the reader along on some of his most surprising and engaging intellectual adventures. From science consulting for a Hollywood mov...

The Wolfram Language represents a major advance in programming languages that makes leading-edge computation accessible to everyone. Unique in its approach of building in vast knowledge and automation, the Wolfram Language scales from a single line of easy-to-understand, interactive code to million-line production systems. This book provides an ele...

This book of thoroughly engaging essays from one of today's most prodigious innovators provides a uniquely personal perspective on the lives and achievements of a selection of intriguing figures from the history of science and technology. Weaving together his immersive interest in people and history with insights gathered from his own experiences,...

The Wolfram Language represents a major advance in programming languages that makes leading-edge computation accessible to everyone. Unique in its approach of building in vast knowledge and automation, the Wolfram Language scales from a single line of easy-to-understand interactive code to million-line production systems. This book provides an ele...

A query in an imprecise syntax is received, and a plurality of assumptions relating to the meaning of the query are determined. One of the assumptions is chosen from the determined plurality of assumptions, and a first response to the query is generated based on the chosen assumption. A user interface mechanism is provided to permit a user to chang...

This talk was given 10 years ago. What it says still stands, but there is now quite a bit more to say. In the 10 years that have passed, much has been done in the exploration of the computational universe, both theoretically, and in applications, particularly in technology. (One notable result, already suspected a decade ago, is a proof of the simp...

In this paper we describe a combination of ideas to improve incremental signature-based Gröbner basis algorithms which have a big impact on their performance. Besides explaining how to combine already known optimizations to achieve more efficient ...

An answer engine determines an entity related to a query from a user in a system designed for computation, search, or other activities involving the processing of queries. The system determines whether an entity page for the entity is stored in an answers database and, if so, retrieves and displays the stored entity page. If there is no stored enti...

In a method for presenting information, an interactive document, for example, may be created that includes both an input expression and a result generated by evaluating the input expression, and where the input expression can be selectively hidden. This may permit “decluttering” of the document so that it may be easier to follow. At the same time,...

Can everything about the universe and its history be computed like the digits of π? Or is it instead uncomputable, like the digits of Chaitin's Ω?

Dedicated to Gregory Chaitin on the occasion of his sixtieth birthday, these remarks attempt to capture some of the kinds of topics we have discussed over the course of many enjoyable hours and days during the past twenty-five years.

Cellular automata are simple mathematical systems that exhibit very complicated behaviour. They can be considered as discrete dynamical systems or as computational systems. Progress has recently been made in studying several aspects of them. Twenty central problems that remain unsolved are discussed.

Whereas once the world view of the ancient world was shifted by Pythagorean mathematics, scientific knowledge and human perception has been contested in the 21st century by developments in computation. In his ground-breaking book Stephen Wolfram, the British physicist and creator of the Mathematica program, asserted that the complexity of the unive...

1. The Foundations for a New Kind of Science 2. The Crucial Experiment 3. The World of Simple Programs 4. Systems Based on Numbers 5. Two Dimensions and Beyond 6. Starting from Randomness 7. Mechanisms in Programs and Nature 8. Implications for Everyday Systems 9. Fundamental Physics 10. Processes of Perception and Analysis 11. The Notion of Comput...

Two decades in the making, this long-awaited work from one of the world's most respected scientists presents a series of dramatic discoveries never before made public. Starting from a collection of simple computer experiments--illustrated in the book by striking computer graphics--Wolfram shows how their unexpected results force a whole new way of...

Are mathematical equations the best way to model nature? For many years it had been assumed that they were. But in the early 1980s, Stephen Wolfram made the radical proposal that one should instead build models that are based directly on simple computer programs. Wolfram made a detailed study of a class of such models known as cellular automata, an...

Cellular automata are mathematical models for systems containing many identical components with local interactions. These notes describe some of their properties, and discuss applications to condensed matter physics.

(Now part of the Wolfram Documentation Center) The Mathematica Book is one of the most unique examples of software documentation in existence. Highly honored for its depth, presentation, and organization, it is a model for all technical manuals. First written in 1988 as a 750-page companion tutorial to the newly created Mathematica software system...

Continuum equations are derived for the large-scale behavior of a class of cellular automaton models for fluids. The cellular automata are discrete analogues of molecular dynamics, in which particles with discrete velocities populate the links of a fixed array of sites. Kinetic equations for microscopic particle distributions are constructed. Hydro...

A 1-dimensional cellular automaton which generates random sequences is discussed. Each site in the cellular automaton has value 0 or 1, and is updated in parallel according to the rule a′i = ai − 1 XOR (ai OR ai + 1) (a′i = (ai − 1 + ai + ai + 1 + aiai + 1) mod 2). Despite the simplicity of this rule, the time sequences of site values that it yield...

This abstract discusses a stream cipher based on a simple one-dimensional cellular automation. The cellular automaton consists of a circular register with N cells, each having a value a i equal to 0 or 1. The values are updated synchronously in discrete time steps according to the rule $$ a'_i = a_{i - 1} XOR (a_i OR a_{i + 1} )

Randomness and chaos in physical systems are usually ultimately attributed to external noise. But it is argued here that even without such random input, the intrinsic behavior of many nonlinear systems can be computationally so complicated as to seem random in all practical experiments. This effect is suggested as the basic origin of such phenomeno...

Contemporary classics on the the major approaches to emergence found in contemporary philosophy and science, with chapters by such prominent scholars as John Searle, Stephen Weinberg, William Wimsatt, Thomas Schelling, Jaegwon Kim, Daniel Dennett, Herbert Simon, Stephen Wolfram, Jerry Fodor, Philip Anderson, David Chalmers, and others. Emergence, l...

A largely phenomenological study of two-dimensional cellular automata is reported. Qualitative classes of behavior similar to those in one-dimensional cellular automata are found. Growth from simple seeds in two-dimensiona! cellular automata can produce patterns with complicated boundaries, characterized by a variety of growth dimensions. Evolution...

Natural systems from snowflakes to mollusc shells show a great diversity of complex patterns. The origins of such complexity can be investigated through mathematical models termed `cellular automata'. Cellular automata consist of many identical components, each simple., but together capable of complex behaviour. They are analysed both as discrete d...

Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities...

Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages. Many examples are g...

Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to lim...

Casimir energies are calculated for quantized fields in cavities with a variety of forms. Consequences for models of the vacuum state are considered. The possibility of negative mass systems is discussed. Results on energy and entropy of finite quantum systems at non-zero temperature are given.

The behaviour of a charged scalar field in an external electric field is discussed. Vacuum polarization is calculated by explicit summation of modes. Instabilities encountered in the external field approximation are absent when back reaction effects are included through a self-consistent semiclassical procedure.

Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the...

A QCD model for e +e - annihilation is presented, and its consequences are compared with experimental data. The model involves production of a shower of partons described by a simple approximation to QCD perturbation theory, and decay of colour singlet clusters of produced partons into hadrons through a simple phase space process. The model reprodu...

Several theoretical principles suggest the existence of large numbers of very massive particles. Such particles have negligible effect in the present universe, but may have been important in the very early universe. It is shown that under some circumstances their presence could completely change the equation of state and expansion rate of the very...

Features of grand unified gauge models relevant to cosmology are discussed. Several SU(5) and SO(10) models are considered in detail. Boltzmann transport equation methods are used to calculate the development of baryon asymmetry in the early universe. Comparison with observation places constraints on possible grand unified models.

Methods for complete calculation of cosmological baryon-number generation in the hot big-bang early universe are outlined and are applied to several SU(5) models. Effects of several baryon-number-nonconserving bosons and the presence of nonthermalizing modes are treated.

by Stephen Wolfram with Chris A. Cole Contributions from Geoffrey C. Fox, Jeffrey M. Greif, Eric D. Mjolsness, Larry J. Romans, Timothy Shaw, and Anthony E. Terrano (California Institute of Technology, 1981; Inference Corporation, 1983) https://www.stephenwolfram.com/publications/smp-symbolic-manipulation-program/

The masses and lifetimes of very heavy weakly interacting fermions which appear in many grand unified gauge models are constrained by the requirement that their decays in the hot big bang early universe should not generate excessive entropy which would dilute nB/nγ below its observed value.

SMP is a new general-purpose symbolic manipulation computer program which has been developed during the past year by the authors, with help from G.C. Fox, J.M. Greif, E.D. Mjolsness, L.J. Romans, T. Shaw and A.E. Terrano. The primary motivation for the construction of the program was the necessity of performing very complicated algebraic manipulati...

The generation of an excess of baryons over antibaryons in the very early universe due to CP-and B-violating interactions is described. The Boltzmann equation is used to perform detailed calculations of the time development of such an excess in several simple illustrative models. Complications encountered in applications of the results to specific...

Correlations between the energies incident on two or three detectors arounde + e – annihilation events are considered as a probe of the QCD structure of the events. Practical methods for deducing two-detector energy correlations (which give the mean product of energies incident on two detectors as a function of their angular separation) from measur...

Gauge theories for weak interactions which employ the Higgs mechanism for spontaneous symmetry breakdown imply that there should exist a large vacuum energy associated with the Higgs scalar field condensate. A cosmological term in Einstein's field equations can be arranged to remove the unobserved gravitational effect of this vacuum energy in the p...

A Monte Carlo model for the development of parton jets in QCD is described. Explicit low-order calculations are supplemented by leading logarithmic approximations for higher orders.

The development of an excess of baryons over antibaryons due to CP and baryon number violating reactions during the very early stages of the big bang is calculated in simple models using the Boltzmann equation. We show that it is necessary to solve the coupled Boltzmann equations in order to determine the final baryon number in any specific model....

The structure of hadronic final states in deep inelastic scattering expected from QCD is analyzed in terms of the shape parameters Hl and Cl. We find that the effects of the fragmentation of quarks and gluons into hadrons are typically governed by . For √s≳30 GeV, the distributions of events in H2 and C2 should allow a test of the perturbative QCD...

The electromagnetic-radiative-correction factors to deep-inelastic neutrino-nucleon scattering are estimated using the parton-model formulas of Kiskis. Tables and approximate formulas are given for the benefit of experimenters wishing to estimate their own corrections. The corrections tend to be less than about 10%, and vary only slowly with energy...

DOI:https://doi.org/10.1103/PhysRevD.20.821

It is pointed out that weak interactions should admix a small parity-violating E1 component into the M1 electromagnetic decay Σ0→Λγ.

We present a new class of observables which distinguish events containing two or three hadron jets from those containing a larger number. These observables, which essentially measure the coplanary of events, are calculable in QCD perturbation theory. Their use should allow the mechanism of decay to be determined.

The standard model of the early universe is used to estimate the present abundances of possible absolutely-stable hadrons or charged leptons more massive than the proton. It is found that experimental limits on their present abundances indicate that no such particles exist with masses below about 16 GeV/c2. Forthcoming experiments could increase th...

Various conditions necessary for the self-consistency of the Weinberg-Salam model are used to place constraints on fermion and Higgs Boson masses. We find that spontaneous symmetry breakdown cannot generate fermion masses in excess of about 300 GeV.

We present a complete set of rotationally invariant observables ( Hl) which characterizes the 'shapes' of final states in e +e - annihilation. They are infrared stable when calculated in QCD perturbation theory. We compare the 'shapes' of final states from the processes e+e- → qoverlineq, e+e- → qoverlineq( G) and from the three-gluon decays of hea...

We calculate the production of heavy charged or strongly interacting particles by cosmic rays and find that any sufficiently light stable ones should be detectable in static terrestrial searches. Stable or nearly stable heavy hadrons would also be visible with a distinct signature in delayed-particle cosmic-ray experiments.

This paper treats several disjointed topics in the study of event shapes in e+ e- annihilation to hadrons. First, we describe the calculation of the shape parameters expectation(Hl) in QCD perturbation theory. We show that they suffer O(m/SQRT(s)) mass corrections, discuss smearing required near thresholds and consider some effects of hadron format...

We present a set of rotationally invariant observables which characterizes the "shapes" of events, and is calculable in quantum-chromodynamics perturbation theory for final states consisting of quarks and gluons (G). We include the effects of fragmentation to hadrons in comparing the shapes of events from the processes e+e--->qq¯, e+e--->qq¯G, and...

The associated production of hadrons containing heavy quarks is studied in the framework of a model based on quark-gluon color gauge field theory [quantum chromodynamics (QCD)]. We assume that the dominant mechanism for the production of heavy quarks in real and virtual photon beams is γ(Q2)V-->cc¯ where V denotes a vector gluon and c an arbitrary...

Arguments are given that quark and gluon momentum distributions generated by asymptotic freedom formulae from distributions which are positive at Q2 = Q02 remain positive for Q2 ⩾ Q02. For Q2 < Q02 they inevitably become negative. Momentum distributions at Q2 = Q02 should be chosen so that this occurs only for very small Q2. This constraints the pa...

We study the angular distribution of muons in the dimuon rest frame from pp→μμ̅ X at high energy and large dimuon mass. Including smearing due to quark transverse momenta, we show that the Drell-Yan model predicts a polar angular distribution in the t-channel helicity frame of 1+αcos2θ with α≳0.8. Experimental deviations from this prediction would...

An analysis is presented of the single inclusive lepton spectra expected from the decay of charmed hadrons or heavier particles with new flavors produced in e+e− annihilation. Extremely soft spectra are found, in contrast to the heavy lepton signal. Good detection efficiency at low momentum will therefore continue to be important in future experime...

A new form of high energy electron-hadron coupling is examined with reference to the experimental data. The electron is taken to have a neutral vector gluon cloud with a radius ~ 10-18 m. This is shown to be consistent with measurements on e+e- -+ e+e- and 9.-2. At low energies, only photons couple to the gluons, but at higher energies 'evaporation...

La littérature sur les automates cellulaires est immense et les ressources Internet qui y sont consacrées sont légions. L'objectif ici est beaucoup plus limité. Ce site se voulant voué aux profanes, je me contenterai d'apporter des éléments de réponse aux deux questions essentielles que se pose toute personne découvrant les auto- mates cellulaires,...

Some approaches to the study of complex systems are outlined. They are encompassed by an emerging field of science concerned with the general analysis of complexity. Throughout the natural and artificial world one observes phenomena of great complexity. Yet research in physics and to some extent biology and other fields has shown that the basic com...

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