Sobolev Institute of Mathematics

Universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy. The price for universality turns out to be acceptably small.

The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc.

The status of the research on muon colliders is discussed and plans are outlined for future theoretical and experimental studies. Besides work on the parameters of a 3--4 and 0.5 TeV center-of-mass (COM) energy collider, many studies are now concentrating on a machine near 0.1 TeV (COM) that could be a factory for the $s$-channel production of Higgs particles. We discuss the research on the various components in such muon colliders, starting from the proton accelerator needed to generate pions from a heavy-$Z$ target and proceeding through the phase rotation and decay ($\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\mu}{\ensuremath{\nu}}_{\ensuremath{\mu}}$) channel, muon cooling, acceleration, storage in a collider ring, and the collider detector. We also present theoretical and experimental R plans for the next several years that should lead to a better understanding of the design and feasibility issues for all of the components. This report is an update of the progress on the research and development since the feasibility study of muon colliders presented at the Snowmass '96 Workshop [R. B. Palmer, A. Sessler, and A. Tollestrup, Proceedings of the 1996 DPF/DPB Summer Study on High-Energy Physics (Stanford Linear Accelerator Center, Menlo Park, CA, 1997)].

A characterization of tame automorphisms of the algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper F left-bracket x 1 comma x 2 comma x 3 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A=F[x_1,x_2,x_3]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of polynomials in three variables over a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained. In particular, it is proved that the well-known Nagata automorphism is wild. It is also proved that the tame and the wild automorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are algorithmically recognizable.

We obtain tree-level unitarity constraints for the most general Two-Higgs-Doublet Model (2HDM) with explicit $CP$-violation. We briefly discuss correspondence between possible violation of tree-level unitarity limitation and physical content of the theory.

ABSTRACT In this paper we prove in dimension n ⪆ 3 that knowledge of the Cauchy data for the Schrödinger equation measured on particular subsets of the boundary determines uniquely the potential.

In this paper we prove that the Cauchy data for the Schrödinger equation in the two-dimensional case determines a potential from L p (for p &gt; 2) uniquely. We also obtain a linear inversion formula for smooth potentials.

The Higgs potential of 2HDM keeps its generic form under the group of transformation $GL(2,C)$, which is larger than the usually considered reparametrization group $SU(2)$. This reparametrization symmetry induces the Minkowski space structure in the orbit space of 2HDM. Exploiting this property, we present a geometric analysis of the number and properties of stationary points of the most general 2HDM potential. In particular, we prove that charge-breaking and neutral vacua never coexist in 2HDM and establish conditions when the most general explicitly $CP$-conserving Higgs potential has spontaneously $CP$-violating minima. We also define the prototypical model of a given 2HDM, which has six free parameters less than the original one but still contains all the essential physics. Our analysis avoids manipulation with high-order algebraic equations.

Translocations and gross deletions are important causes of both cancer and inherited disease. Such gene rearrangements are nonrandomly distributed in the human genome as a consequence of selection for growth advantage and/or the inherent potential of some DNA sequences to be frequently involved in breakage and recombination. Using the Gross Rearrangement Breakpoint Database [GRaBD; www.uwcm.ac.uk/uwcm/mg/grabd/grabd.html] (containing 397 germ-line and somatic DNA breakpoint junction sequences derived from 219 different rearrangements underlying human inherited disease and cancer), we have analyzed the sequence context of translocation and deletion breakpoints in a search for general characteristics that might have rendered these sequences prone to rearrangement. The oligonucleotide composition of breakpoint junctions and a set of reference sequences, matched for length and genomic location, were compared with respect to their nucleotide composition. Deletion breakpoints were found to be AT-rich whereas by comparison, translocation breakpoints were GC-rich. Alternating purine-pyrimidine sequences were found to be significantly over-represented in the vicinity of deletion breakpoints while polypyrimidine tracts were over-represented at translocation breakpoints. A number of recombination-associated motifs were found to be over-represented at translocation breakpoints (including DNA polymerase pause sites/frameshift hotspots, immunoglobulin heavy chain class switch sites, heptamer/nonamer V(D)J recombination signal sequences, translin binding sites, and the chi element) but, with the exception of the translin-binding site and immunoglobulin heavy chain class switch sites, none of these motifs were over-represented at deletion breakpoints. Alu sequences were found to span both breakpoints in seven cases of gross deletion that may thus be inferred to have arisen by homologous recombination. Our results are therefore consistent with a role for homologous unequal recombination in deletion mutagenesis and a role for nonhomologous recombination in the generation of translocations.

We consider the open shop, job shop, and flow shop scheduling problems with integral processing times. We give polynomial-time algorithms to determine if an instance has a schedule of length at most 3, and show that deciding if there is a schedule of length at most 4 is 𝒩𝒫-complete. The latter result implies that, unless 𝒫 = 𝒩𝒫, there does not exist a polynomial-time approximation algorithm for any of these problems that constructs a schedule with length guaranteed to be strictly less than 5/4 times the optimal length. This work constitutes the first nontrivial theoretical evidence that shop scheduling problems are hard to solve even approximately.

We consider the problem of scheduling n jobs with release dates on m machines so as to minimize their average weighted completion time. We present the first known polynomial time approximation schemes for several variants of this problem. Our results include PTASs for the case of identical parallel machines and a constant number of unrelated machines with and without preemption allowed. Our schemes are efficient: for all variants the running time for /spl alpha/(1+/spl epsiv/) approximation is of the form f(1//spl epsiv/, m)poly(n).

We use the invariance of a physical picture under a change of Lagrangian, the reparameterization invariance in the space of Lagrangians and its particular case---the rephrasing invariance---for analysis of the two-Higgs-doublet extension of the standard model. We found that some parameters of theory like $\mathrm{tan}\ensuremath{\beta}$ are reparameterization dependent and therefore cannot be fundamental. We use the ${Z}_{2}$ symmetry of the Lagrangian, which prevents a ${\ensuremath{\phi}}_{1}\ensuremath{\leftrightarrow}{\ensuremath{\phi}}_{2}$ transition, and the different levels of its violation, soft and hard, to describe the physical content of the model. In general, the broken ${Z}_{2}$ symmetry allows for a $CP$ violation in the physical Higgs sector. We argue that the two-Higgs-doublet model with a soft breaking of ${Z}_{2}$ symmetry is a natural model in the description of electroweak symmetry breaking. To simplify the analysis, we choose among different forms of Lagrangian describing the same physical reality a specific one, in which the vacuum expectation values of both Higgs fields are real. A possible $CP$ violation in the Higgs sector is described by using a two-step procedure with the first step identical to a diagonalization of the mass matrix for $CP$-even fields in the $CP$-conserving case. We find a very simple, necessary, and sufficient condition for a $CP$ violation in the Higgs sector. We determine the range of parameters for which $CP$ violation and flavor-changing neutral current effects are naturally small---it corresponds to a small dimensionless mass parameter $\ensuremath{\nu}=\mathrm{Re}{m}_{12}^{2}/(2{v}_{1}{v}_{2})$. We show that for small $\ensuremath{\nu}$ some Higgs bosons can be heavy---with mass up to about 0.6 TeV---without violating of the unitarity constraints. If $\ensuremath{\nu}$ is large, all Higgs bosons except one can be arbitrarily heavy. We discuss, in particular, main features of this case, which corresponds for $\ensuremath{\nu}\ensuremath{\rightarrow}\ensuremath{\infty}$ to a decoupling of heavy Higgs bosons. In the model II for Yukawa interactions we obtain the set of relations among the couplings to gauge bosons and to fermions which allows us to analyze different physical situations (including $CP$ violation) in terms of these very couplings, instead of the parameters of Lagrangian.

We continue to explore the consequences of the recently discovered Minkowski space structure of the Higgs potential in the two-Higgs-doublet model. Here, we focus on the vacuum properties. The search for extrema of the Higgs potential is reformulated in terms of 3-quadrics in the $3+1$-dimensional Minkowski space. We prove that 2HDM cannot have more than two local minima in the orbit space and that a twice-degenerate minimum can arise only via spontaneous violation of a discrete symmetry of the Higgs potential. Investigating topology of the 3-quadrics, we give concise criteria for existence of noncontractible paths in the Higgs orbit space. We also study explicit symmetries of the Higgs potential/Lagrangian and their spontaneous violation from a wider perspective than usual.

We describe the status of our effort to realize a first neutrino factory and the progress made in understanding the problems associated with the collection and cooling of muons towards that end. We summarize the physics that can be done with neutrino factories as well as with intense cold beams of muons. The physics potential of muon colliders is reviewed, both as Higgs factories and compact high-energy lepton colliders. The status and time scale of our research and development effort is reviewed as well as the latest designs in cooling channels including the promise of ring coolers in achieving longitudinal and transverse cooling simultaneously. We detail the efforts being made to mount an international cooling experiment to demonstrate the ionization cooling of muons.

This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics recently proposed by Peshkov and Romenski [110], further denoted as HPR model. In that framework, the viscous stresses are computed from the so-called distortion tensor A, which is one of the primary state variables in the proposed first order system. A very important key feature of the HPR model is its ability to describe at the same time the behavior of inviscid and viscous compressible Newtonian and non-Newtonian fluids with heat conduction, as well as the behavior of elastic and visco-plastic solids. Actually, the model treats viscous and inviscid fluids as generalized visco-plastic solids. This is achieved via a stiff source term that accounts for strain relaxation in the evolution equations of A. Also heat conduction is included via a first order hyperbolic system for the thermal impulse, from which the heat flux is computed. The governing PDE system is hyperbolic and fully consistent with the first and the second principle of thermodynamics. It is also fundamentally different from first order Maxwell–Cattaneo-type relaxation models based on extended irreversible thermodynamics. The HPR model represents therefore a novel and unified description of continuum mechanics, which applies at the same time to fluid mechanics and solid mechanics. In this paper, the direct connection between the HPR model and the classical hyperbolic–parabolic Navier–Stokes–Fourier theory is established for the first time via a formal asymptotic analysis in the stiff relaxation limit. From a numerical point of view, the governing partial differential equations are very challenging, since they form a large nonlinear hyperbolic PDE system that includes stiff source terms and non-conservative products. We apply the successful family of one-step ADER–WENO finite volume (FV) and ADER discontinuous Galerkin (DG) finite element schemes to the HPR model in the stiff relaxation limit, and compare the numerical results with exact or numerical reference solutions obtained for the Euler and Navier–Stokes equations. Numerical convergence results are also provided. To show the universality of the HPR model, the paper is rounded-off with an application to wave propagation in elastic solids, for which one only needs to switch off the strain relaxation source term in the governing PDE system. We provide various examples showing that for the purpose of flow visualization, the distortion tensor A seems to be particularly useful.

A comprehensive review of physics at an [Formula: see text] linear collider in the energy range of [Formula: see text] GeV-3 TeV is presented in view of recent and expected LHC results, experiments from low-energy as well as astroparticle physics. The report focusses in particular on Higgs-boson, top-quark and electroweak precision physics, but also discusses several models of beyond the standard model physics such as supersymmetry, little Higgs models and extra gauge bosons. The connection to cosmology has been analysed as well.

High energy photon colliders (γγ,γe) are based on e - e - linear colliders where high energy photons are produced using Compton scattering of laser light on high energy electrons just before the interaction point. This paper is a part of the Technical Design Report of the linear collider TESLA. 1 Physics program, possible parameters and some technical aspects of the photon collider at TESLA are discussed.

Understanding how the time complexity of evolutionary algorithms (EAs) depend on their parameter settings and characteristics of fitness landscapes is a fundamental problem in evolutionary computation. Most rigorous results were derived using a handful of key analytic techniques, including drift analysis. However, since few of these techniques apply effortlessly to population-based EAs, most time complexity results concern simple EAs, such as the (1+1) EA. We present the level-based theorem, a new technique tailored to population-based processes. It applies to any nonelitist process where offspring are sampled independently from a distribution depending only on the current population. Given conditions on this distribution, our technique provides upper bounds on the expected time until the process reaches a target state. The technique is demonstrated on pseudo-Boolean functions, the sorting problem, and approximation of optimal solutions in combinatorial optimization. The conditions of the theorem are often straightforward to verify, even for genetic algorithms and estimation of distribution algorithms which were considered highly nontrivial to analyze. The proofs for the example applications are available in the supplementary materials. Finally, we prove that the theorem is nearly optimal for the processes considered. Given the information the theorem requires about the process, a much tighter bound cannot be proved.

This book focuses on the asymptotic behaviour of the probabilities of large deviations of the trajectories of random walks with 'heavy-tailed' (in particular, regularly varying, sub- and semiexponential) jump distributions. Large deviation probabilities are of great interest in numerous applied areas, typical examples being ruin probabilities in risk theory, error probabilities in mathematical statistics, and buffer-overflow probabilities in queueing theory. The classical large deviation theory, developed for distributions decaying exponentially fast (or even faster) at infinity, mostly uses analytical methods. If the fast decay condition fails, which is the case in many important applied problems, then direct probabilistic methods usually prove to be efficient. This monograph presents a unified and systematic exposition of the large deviation theory for heavy-tailed random walks. Most of the results presented in the book are appearing in a monograph for the first time. Many of them were obtained by the authors.

We assume that the current state of the Universe can be described by the inert doublet model, containing two scalar doublets, one of which is responsible for electroweak-symmetry breaking and masses of particles and the second one having no couplings to fermions and being responsible for dark matter. We consider possible evolutions of the Universe to this state during cooling down of the Universe after inflation. We found that in the past the Universe could pass through phase states having no dark matter candidate. In the evolution via such states, in addition to a possible electroweak-symmetry breaking phase transition (second order), the Universe sustained one first-order phase transition or two phase transitions of the second order.