Integer Length Research Articles

Superimposed particles in 1D ground states

For a class of nonnegative, range-1 pair potentials in one-dimensional continuous space we prove that any classical ground state of lower density ⩾1 is a tower-lattice, i.e. a lattice formed by towers of particles the heights of which can differ only by 1, and the lattice constant is 1. The potential may be flat or may have a cusp at the origin; it can be continuous, but its derivative has a jump at 1. The result is valid on finite intervals or rings of integer length and on the whole line.

Mutually independent bipanconnected property of hypercube

A graph is denoted by G with the vertex set V( G) and the edge set E( G). A path P = 〈 v 0, v 1, … , v m 〉 is a sequence of adjacent vertices. Two paths with equal length P 1 = 〈 u 1, u 2, … , u m 〉 and P 2 = 〈 v 1, v 2, … , v m 〉 from a to b are independent if u 1 = v 1 = a, u m = v m = b, and u i ≠ v i for 2 ⩽ i ⩽ m − 1. Paths with equal length { P i } i = 1 n from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d G ( u, v) ⩽ l ⩽ ∣ V( G) − 1∣ with ( l − d G ( u, v)) being even. We say that the pair of vertices u, v is ( m, l)- mutually independent bipanconnected if there exist m mutually independent paths P i l i = 1 m with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d Q n ( u , v ) ⩾ n - 1 , is ( n − 1, l)-mutually independent bipanconnected for every l , d Q n ( u , v ) ⩽ l ⩽ | V ( Q n ) - 1 | with ( l - d Q n ( u , v ) ) being even. As for d Q n ( u , v ) ⩽ n - 2 , it is also ( n − 1, l)-mutually independent bipanconnected if l ⩾ d Q n ( u , v ) + 2 , and is only ( l, l)-mutually independent bipanconnected if l = d Q n ( u , v ) .

Separators of fat points in Pn

Abstract In this paper we extend the definition of a separator of a point P in P n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m 1 P 1 + ⋯ + m i P i + ⋯ + m s P s ⊆ P n and Z ′ = m 1 P 1 + ⋯ + ( m i − 1 ) P i + ⋯ + m s P s . We associate to P i a tuple of positive integers of length ν = deg Z − deg Z ′ . We call this tuple the degree of the minimal separators of P i of multiplicity m i , and we denote it by deg Z ( P i ) = ( d 1 , … , d ν ) . We show that if one knows deg Z ( P i ) and the Hilbert function of Z, one will also know the Hilbert function of Z ′ . We also show that the entries of deg Z ( P i ) are related to the shifts in the last syzygy module of I Z . Both results generalize well-known results about reduced sets of points and their separators.

Pythagorean quantization, action(s) and the arrow of time

Searching for the first well-documented attempts of introducing some kind of "quantization" into the description of nature inevitably leads to the ancient Greeks, in particular Plato and Pythagoras. The question of finding the so-called Pythagorean triples, i.e., right-angled triangles with integer length of all three sides, is, surprisingly, connected with complex nonlinear Riccati equations that occur in time-dependent quantum mechanics. The complex Riccati equation together with the usual Newtonian equation of the system, leads to a dynamical invariant with the dimension of an action. The relation between this invariant and a conserved "angular momentum" for the motion in the complex plane will be determined. The "Pythagorean quantization" shows similarities with the quantum Hall effect and leads to an interpretation of Sommerfeld's fine structure constant that involves another quantum of action, the "least Coulombic action" e2/c. Since natural evolution is characterized by irreversibility and dissipation, the question of how these aspects can be incorporated into a quantum mechanical description arises. Two effective approaches that also both possess a dynamical invariant (like the one mentioned above) will be discussed. One uses an explicitly time-dependent (linear) Hamiltonian, whereas the other leads to a nonlinear Schrödinger equation with complex logarithmic nonlinearity. Both approaches can be transformed into each other via a non-unitary transformation that involves Schrödinger's original definition of a (complex) action via the wave function.

PACKING SHELVES WITH ITEMS THAT DIVIDE THE SHELVES' LENGTH: A CASE OF A UNIVERSAL NUMBER PARTITION PROBLEM

This note examines the likelihood of packing two identical one dimensional shelves of integer length L by items whose individual lengths are divisors of L, given that their combined length sums-up to 2L. We compute the number of packing failures and packing successes for integer shelve lengths L, 1 ≤ L ≤ 1000, by implementing a dynamic programming scheme using a problem specific "boundedness property". The computational results indicate that the likelihood of a packing failure is very rare. We observe that the existence of packing failures is tied to the number of divisors of L and prove that the number of divisors has to be at least 8 for a packing failure to exist.

Scheduling of coupled tasks with unit processing times

The coupled tasks scheduling problem was originally introduced for modeling complex radar devices. It is still used for controlling such devices and applied in similar applications. This paper considers a problem of coupled tasks scheduling on a single processor, under the assumptions that all processing times are equal to 1, the gap has exact integer length L and the precedence constraints are strict. We prove that the general problem, when L is part of the input and the precedence constraints graph is a general graph, is NP-hard in the strong sense. We also show that the special case when L=2 and the precedence constraints graph is an in-tree or an out-tree, can be solved in O(n) time.

Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length

We consider pyramids made of one-dimensional pieces of fixed integer length $a$ and which may have pairwise overlaps of integer length from $1$ to $a$. We give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between so-called right (or left) pyramids and $a$-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.

The Littlewood—Paley—Rubio de Francia property of a Banach space for the case of equal intervals

Let X be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of X-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents p ≥ 2 if and only if the space X is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.

Straight line embeddings of cubic planar graphs with integer edge lengths

AbstractWe prove that every simple cubic planar graph admits a planar embedding such that each edge is embedded as a straight line segment of integer length. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:270‐274, 2008

Identification and signatures based on NP-hard problems of indefinite quadratic forms

We prove NP -hardness of equivalence and representation problems of quadratic forms under probabilistic reductions, in particular for indefinite, ternary quadratic forms with integer coefficients. We present identifications and signatures based on these hard problems. The bit complexity of signature generation and verification is quadratic using integers of bit length 150.

Primary calibration of coiled brachytherapy sources

Coiled brachytherapy sources have been developed by RadioMed Corporation for use as low‐dose‐rate (LDR) interstitial implants. The coiled sources are provided in integer lengths from 1 to and address many common issues seen with traditional LDR brachytherapy sources. The current standard for determining the air‐kerma strength of low‐energy LDR brachytherapy sources is the National Institute of Standards and Technology's Wide‐Angle Free‐Air Chamber (NIST WAFAC). Due to geometric limitations, however, the NIST WAFAC is unable to determine the of sources longer than . This project utilized the University of Wisconsin's Variable‐Aperture Free‐Air Chamber (UW VAFAC) to determine the of the longer coiled sources. The UW VAFAC has shown agreement in values of length coils to within 1% of those determined with the NIST WAFAC, but the UW VAFAC does not share the same geometric limitations as the NIST WAFAC. A new source holder was constructed to hold the coiled sources in place during measurements with the UW VAFAC. Correction factors for the increased length of the sources have been determined and applied to the measurements. Using the new source holder and corrections, the of 3 and coiled sources has been determined. Corrected UW VAFAC data and ionization current measurements from well chambers have been used to determine calibration coefficients for use in the measurement of 3 and coiled sources in well chambers. Thus, the UW VAFAC has provided the first transferable, primary measurement of low‐energy LDR brachytherapy sources with lengths greater than .

Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths

AbstractWe study the complexity of two inverse shortest paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph D = (V, A), the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. In the first problem we seek for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is 𝒩𝒫‐hard to approximate the minimal longest path length within a factor less than 8/7 or the minimal longest arc length within a factor less than 9/8. This answers the (previously) open question whether these problems are 𝒩𝒫‐hard or not. We also present a simple algorithm that achieves an 𝒪(∣V∣)‐approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is 𝒩𝒫‐hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS‐IS, or RIP. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 29–36 2007

Application of the Ramanujan Fourier Transform for the Analysis of Secondary Structure Content in Amino Acid Sequences

A novel method is presented for the investigation of protein properties of sequences using Ramanujan Fourier Transform (RFT). The new methodology involves the preprocessing of protein sequence data by numerically encoding it and then applying the RFT. The RFT is based on projecting the obtained numerical series on a set of basis functions constituted by Ramanujan sums (RS). In RS components, periodicities of finite integer length, rather than frequency, (as in classical harmonic analysis) are considered. The potential of the new approach is documented by a few examples in the analysis of hydrophobic profiles of proteins in two classes including abundance of alpha-helices (group A) or beta-strands (group B). Different patterns are provided as evidence. RFT can be used to characterize the structural properties of proteins and integrate complementary information provided by other signal processing transforms.

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u , v ∈ V ( G ) , a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if d C ( u , v ) = max { d C ( x , y ) | x , y ∈ V ( C ) } . A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271–278] if it contains a cycle of every length from 3 to | V ( G ) | inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181–187] (respectively, balanced pancyclic) if for each pair of vertices u , v ∈ V ( G ) , it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max { 2 d G ( u , v ) , 3 } ⩽ l ⩽ | V ( G ) | . Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251] proved that the n-dimensional Augmented cube, AQ n , is pancyclic in the sense that a cycle of length l exists, 3 ⩽ l ⩽ | V ( AQ n ) | . In this paper, we study two new pancyclic properties and show that AQ n is geodesic pancyclic and balanced pancyclic for n ⩾ 2 .

Realizing partitions respecting full and partial order information

For n ∈ N , we consider the problem of partitioning the interval [ 0 , n ) into k subintervals of positive integer lengths ℓ 1 , … , ℓ k such that the lengths satisfy a set of simple constraints of the form ℓ i ⋄ i j ℓ j where ⋄ i j is one of <, >, or =. In the full information case, ⋄ i j is given for all 1 ⩽ i , j ⩽ k . In the sequential information case, ⋄ i j is given for all 1 < i < k and j = i ± 1 . That is, only the relations between the lengths of consecutive intervals are specified. The cyclic information case is an extension of the sequential information case in which the relationship ⋄ 1 k between ℓ 1 and ℓ k is also given. We show that all three versions of the problem can be solved in time polynomial in k and log n .

On binary signed digit representations of integers

Applications of signed digit representations of an integer include computer arithmetic, cryptography, and digital signal processing. An integer of length n bits can have several binary signed digit (BSD) representations and their number depends on its value and varies with its length. In this paper, we present an algorithm that calculates the exact number of BSDR of an integer of a certain length. We formulate the integer that has the maximum number of BSDR among all integers of the same length. We also present an algorithm to generate a random BSD representation for an integer starting from the most significant end and its modified version which generates all possible BSDR. We show how the number of BSD representations of k increases as we prepend 0s to its binary representation.

Variable quality compression of fluid dynamical data sets using a 3‐D DCT technique

In this paper we present a data compression scheme that is especially suited for the compression of data sets resulting from computational fluid dynamics (CFD). By adopting the concept of the JPEG image compression standard and extending the approach of Schmalzl (2003), we employ a three‐dimensional discrete cosine transform of the data. The resulting frequency components are rearranged, quantized, and, finally, stored using standard variable length integer codes. The compression ratio and also the introduced loss of accuracy can be adjusted over a wide range by means of two compression parameters to give the desired compression profile. Using the proposed technique, compression ratios of more than 60:1 are possible with a mean error of the compressed data of less than 0.05%.

Farmer Ted Goes 3D

In his recent paper entitled Ted Goes Natural [2], Greg Martin discusses the plight of Farmer Ted, who wants to minimize the cost of chicken wire to enclose a rectangular coop with a base of 190 square feet. However, Farmer Ted has difficulty purchasing 4,j56 feet of chicken wire because he is only able to purchase wire in integer lengths. After much deliberation, he decides to build an 11 foot x 17 foot coop (187 square feet), which has the best cost-efficiency of any possible coop with integer side lengths and area 190 square feet. Here we add another dimension to the story of Farmer Ted. In recent years, Farmer Ted has helped numerous neighbors efficiently build chicken coops. Then one local farmer asked a question that intrigued him: The farmer wants to build an animal cage with a volume of up to 50 cubic feet. What is the most cost-effective way to do this with integer side lengths? Farmer Ted had begun to think in 3D. Before analyzing the three-dimensional problem, we should briefly review some of Martin's work. He starts with the basic calculus optimization problem of finding the minimum perimeter of a rectangle given a fixed area. As many students know, the optimal shape is a square. Martin then offers this variation: Given a positive integer N, what are the dimensions of the rectangle with integer side lengths and area at most N whose area-to-perimeter ratio is largest among all such rectangles? In order to solve this problem, Martin makes the following definitions: Let s(n) = min (c d) where cd =n and c, d E N. Equivalently, s(n) = min (d nld) where d I n. This gives the minimum semiperimeter, or half of the actual perimeter, of a rectangle with area n with integer sides. Define F (n) = n /s (n), which is the maximum ratio of area to semiperimeter for a given n. (Using this expression, which is twice as large as the maximum ratio of area to perimeter, is more aesthetically pleasing and does not change the analysis.) Farmer Ted prefers to enclose 187 ft.2 rather than 190 ft.2 because F(187) = 187/28 > F(190) = 190/29. This leads Martin to identify the set

Visualizing integer length integer vectors with J and POV-Ray

Integer length integer vectors correspond to solutions of Diophantine equations where a sum of squares is a square. In two dimensions, this equation is a 2 + b 2 = c 2 , and solutions in positive integers correspond to Pythagorean Triples (or integer length integer vectors &lt; a , b &gt; in Z 2 ). In three dimensions, this equation is x 2 + y 2 + z 2 = r 2 , and integer solutions correspond to integer length integer vectors &lt; x , y , z &gt; in Z 3 . We investigate methods of visualizing twodimensional integer length integer vectors with J 5.04 [6], and then create three-dimensional generalizations with J and POV-Ray 3.6 [7]. This work has been motivated in part by the Perfect Cuboid and Perfect Parallelepiped problems, which are open problems regarding integer length integer vectors [3]. A discussion of these problems, as well as the parameterizations and algebraic structures of integer length integer vectors in Z 2 and Z 3 is available at [2].

On finding cycle bases and fundamental cycle bases with a shortest maximal cycle

An undirected biconnected graph G with nonnegative integer lengths on the edges is given. The problem we consider is that of finding a cycle basis B of G such that the length of the longest cycle included in B is the smallest among all cycle bases of G. We first observe that Horton's algorithm [SIAM J. Comput. 16 (2) (1987) 358–366] provides a fast solution of the problem that extends the one given by Chickering et al. [Inform. Process. Lett. 54 (1995) 55–58] for uniform graphs. On the other hand we show that, if the basis is required to be fundamental, then the problem is NP-hard and cannot be approximated within 2− ϵ, ∀ ϵ>0, even with uniform lengths, unless P=NP. This problem remains NP-hard even restricted to the class of complete graphs; in this case it cannot be approximated within 13/11− ϵ, ∀ ϵ>0, unless P=NP; it is instead approximable within 2 in general, and within 3/2 if the triangle inequality holds.