On the structure of non-full-rank perfect codes

The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $\mu$-components $K_\mu$, where $\mu$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\mu$-components, and new lower bounds on the number of perfect 1-error-correcting $q$-ary codes are presented.Comment: 8 page

On the Identity and Group Problems for Complex Heisenberg Matrices

We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $\mathbb{Z}^{4 \times 4}$ and decidable for $\mathbb{Z}^{2 \times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time

Using the Landsat data archive to assess long-term regional forest dynamics assessment in Eastern Europe, 1985-2012

Abstract. Dramatic political and economic changes in Eastern European countries following the dissolution of the “Eastern Bloc” and the collapse of the Soviet Union greatly affected land-cover and land-use trends. In particular, changes in forest cover dynamics may be attributed to the collapse of the planned economy, agricultural land abandonment, economy liberalization, and market conditions. However, changes in forest cover are hard to quantify given inconsistent forest statistics collected by different countries over the last 30 years. The objective of our research was to consistently quantify forest cover change across Eastern Europe from 1985 until 2012 using the complete Landsat data archive. We developed an algorithm for processing imagery from different Landsat platforms and sensors (TM and ETM+), aggregating these images into a common set of multi-temporal metrics, and mapping annual gross forest cover loss and decadal gross forest cover gain. Our results show that forest cover area increased from 1985 to 2012 by 4.7% across the region. Average annual gross forest cover loss was 0.41% of total forest cover area, with a statistically significant increase from 1985 to 2012. Most forest disturbance recovered fast, with only 12% of the areas of forest loss prior to 1995 not being recovered by 2012. Timber harvesting was the main cause of forest loss. Logging area declined after the collapse of socialism in the late 1980s, increased in the early 2000s, and decreased in most countries after 2007 due to the global economic crisis. By 2012, Central and Baltic Eastern European countries showed higher logging rates compared to their Western neighbours. Comparing our results with official forest cover and change estimates showed agreement in total forest area for year 2010, but with substantial disagreement between Landsat-based and official net forest cover area change. Landsat-based logging areas exhibit strong relationship with reported roundwood production at national scale. Our results allow national and sub-national level analysis of forest cover extent, change, and logging intensity and are available on-line as a baseline for further analyses of forest dynamics and its drivers

On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond

We consider the following variant of the Mortality Problem: given $k\times k$ matrices $A_1, A_2, \dots,A_{t}$, does there exist nonnegative integers $m_1, m_2, \dots,m_t$ such that the product $A_1^{m_1} A_2^{m_2} \cdots A_{t}^{m_{t}}$ is equal to the zero matrix? It is known that this problem is decidable when $t \leq 2$ for matrices over algebraic numbers but becomes undecidable for sufficiently large $t$ and $k$ even for integral matrices. In this paper, we prove the first decidability results for $t>2$. We show as one of our central results that for $t=3$ this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for $t=3$ and $k \leq 3$ for matrices over algebraic numbers and for $t=3$ and $k=4$ for matrices over real algebraic numbers. Another consequence is that the set of triples $(m_1,m_2,m_3)$ for which the equation $A_1^{m_1} A_2^{m_2} A_3^{m_3}$ equals the zero matrix is equal to a finite union of direct products of semilinear sets. For $t=4$ we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular $2 \times 2$ rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations

The membership problem for subsemigroups of GL2(Z) is NP-complete

We show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2x2 matrices from the General Linear Group GL(2,Z) is solvable in NP. We extend this to prove that the membership problem is decidable in NP for GL(2,Z) and for any arbitrary regular expression over matrices from the Special Linear group SL(2,Z). We show that determining if a given finite set of matrices from SL(2,Z) or the modular group PSL(2,Z) generates a group or a free semigroup are decidable in NP. Previous algorithms, shown in 2005 by Choffrut and Karhumäki, were in EXPSPACE. Our algorithm is based on new techniques allowing us to operate on compressed word representations of matrices without explicit expansions. When combined with known NP-hard lower bounds, this proves that the membership problem over GL(2,Z) is NP-complete, and the group problem and the non-freeness problem in SL(2,Z) are NP-complete

Quantum Phase Shift in Chern-Simons Modified Gravity

Using a unified approach of optical-mechanical analogy in a semiclassical formula, we evaluate the effect of Chern-Simons modified gravity on the quantum phase shift of de Broglie waves in neutron interferometry. The phase shift calculated here reveals, in a single equation, a combination of effects coming from Newtonian gravity, inertial forces, Schwarzschild and Chern-Simons modified gravity. However the last two effects, though new, turn out to be too tiny to be observed, and hence only of academic interest at present. The approximations, wherever used, as well as the drawbacks of the non-dynamical approach are clearly indicated.Comment: 16 pages, minor errors corrected. Accepted for publication in Phys. Rev.

Transition from damage to fragmentation in collision of solids

We investigate fracture and fragmentation of solids due to impact at low energies using a two-dimensional dynamical model of granular solids. Simulating collisions of two solid discs we show that, depending on the initial energy, the outcome of a collision process can be classified into two states: a damaged and a fragmented state with a sharp transition in between. We give numerical evidence that the transition point between the two states behaves as a critical point, and we discuss the possible mechanism of the transition.Comment: Revtex, 12 figures included. accepted by Phys. Rev.

Fragmentation of a Circular Disc by Impact on a Frictionless Plate

The break-up of a two-dimensional circular disc by normal and oblique impact on a hard frictionless plate is investigated by molecular dynamics simulations. The disc is composed of numerous unbreakable randomly shaped convex polygons connected together by simple elastic beams that break when bent or stretched beyond a certain limit. It is found that for both normal and oblique impacts the crack patterns are the same and depend solely on the normal component of the impact velocity. Analysing the pattern of breakage, amount of damage, fragment masses and velocities, we show the existence of a critical velocity which separates two regimes of the impact process: below the critical point only a damage cone is formed at the impact site (damage), cleaving of the particle occurs at the critical point, while above the critical velocity the disc breaks into several pieces (fragmentation). In the limit of very high impact velocities the disc suffers complete disintegration (shattering) into many small fragments. In agreement with experimental results, fragment masses are found to follow the Gates-Gaudin-Schuhmann distribution (power law) with an exponent independent of the velocity and angle of impact. The velocity distribution of fragments exhibit an interesting anomalous scaling behavior when changing the impact velocity and the size of the disc.Comment: submitted to J. Phys: Condensed Matter special issue on Granular Medi

Force Distribution in a Granular Medium

We report on systematic measurements of the distribution of normal forces exerted by granular material under uniaxial compression onto the interior surfaces of a confining vessel. Our experiments on three-dimensional, random packings of monodisperse glass beads show that this distribution is nearly uniform for forces below the mean force and decays exponentially for forces greater than the mean. The shape of the distribution and the value of the exponential decay constant are unaffected by changes in the system preparation history or in the boundary conditions. An empirical functional form for the distribution is proposed that provides an excellent fit over the whole force range measured and is also consistent with recent computer simulation data.Comment: 6 pages. For more information, see http://mrsec.uchicago.edu/granula