Integer Length Research Articles

Min-Deviation-Flow in Bi-directed Graphs for T-Mesh Quantization

Subdividing non-conforming T-mesh layouts into conforming quadrangular meshes is a core component of state-of-the-art (re-)meshing methods. Typically, the required constrained assignment of integer lengths to T-Mesh edges is left to generic branch-and-cut solvers, greedy heuristics, or a combination of the two. This either does not scale well with input complexity or delivers suboptimal result quality. We introduce the Minimum-Deviation-Flow Problem in bi-directed networks (Bi-MDF) and demonstrate its use in modeling and efficiently solving a variety of T-Mesh quantization problems. We develop a fast approximate solver as well as an iterative refinement algorithm based on matching in graphs that solves Bi-MDF exactly. Compared to the state-of-the-art QuadWild [Pietroni et al. 2021] implementation on the authors' 300 dataset, our exact solver finishes after only 0.49% (total 17.06s) of their runtime (3491s) and achieves 11% lower energy while an approximation is computed after 0.09% (3.19s) of their runtime at the cost of 24% increased energy. A novel half-arc-based T-Mesh quantization formulation extends the feasible solution space to include previously unattainable quad meshes. The Bi-MDF problem is more general than our application in layout quantization, potentially enabling similar speedups for other optimization problems that fit into the scheme, such as quad mesh refinement.

How to Fold a Triangle: The Connection between the Grashof Fourbar Linkage and Pythagorean Triangles

Summary This paper presents two different methods for folding a triangle onto a flat line by converting it into a Grashof Special Case fourbar linkage. Foldable triangles have many applications, ranging from space structures to collapsible furniture. In the first method, a generic triangle with positive real side lengths is shown to be foldable inward and outward by adding a pin joint at a specific location to one of the sides. The second method shows how to create a collapsible linkage using a Pythagorean triangle; we demonstrate that all four links in the Pythagorean fourbar have integer length and can be easily built using rods with uniformly spaced holes (e.g., LEGO bricks, Unistrut, etc.). Next, the formulas for finding the interior angles of the Pythagorean fourbar are presented, for the purpose of plotting collapsible structures with mathematical software. We conclude with a demonstration of a sample application of Pythagorean fourbars to collapsible arch or tower structures and a surprising proof that some types of collapsible arch structures are unrealizable with Pythagorean fourbars.

Heron triangles with two rational medians and Somos-5 sequences

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E({mathbb {Q}}) with Mordell–Weil group {mathbb {Z}},{times }, {mathbb {Z}}/2{mathbb {Z}}, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel–Roberts–Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

Reconstruction of Talbot self-image using Fourier ptychographic microscopy

Fourier ptychographic microscopy (FPM) is used to merge a sequence of low-resolution Talbot images into a high-resolution image with a large field of view such that both integer and fractional Talbot lengths are clearly resolved. It is also revealed that the diffraction orders present in the dark field Fourier plane images reconstructed based on the Talbot effect i.e. the microscope’s back focal plane images are the ones responsible for rendering FPM amenable for the self-imaging phenomena. These diffraction spots have an arced-shaped diffraction patterns which are consistent with the Talbot image when illuminated with a parabolic wavefront. i.e. Gaussian sums only for the dark field images, and are crucial for FPM to successfully achieve a full complex image recovery. Herein, the Talbot image is treated as a phase-recovery problem, and further analyses are presented by comparing the performance of FPM when the periodic grating Talbot effect is imaged with other standard phase-recovery methods, namely the Fienup and Gercberg-Saxton algorithms.

Mechanical, microstructural and tribological behavior of stir cast Al8011/TiB2/SiC hybrid metal matrix composites

Modern technology requires enriched mechanical and thermal characteristics. In the present scenario, the decent mechanical and thermal properties of hybrid Metal matrix composites (MMCs) make them more demanding in various fields such as automotive, aviation, and structural applications. In this paper, an attempt has been made to fabricate the composite Al8011 + TiB2 + SiC with 9 different weight fractions follows as reinforcement 1(TiB2) by 2%,4%, and 6% with varying reinforcement 2(SiC) percentage by 1%,3%, and 5%. Fabrication is made through the stir casting technique by considering ideal process parameters such as stirring temperature, stirring speed, stirring time, preheating temperature, grain size, and wt.% of particulate reinforcements. With the help of Electron discharge machining (EDM), the specimens for the tests performed were cut according to the ASTM standards. The test results in the present investigation revealed that an increase in mechanical properties has been detected with the addition of reinforcements when compared with individual matrix and reinforcement materials. Wear test analysis illustrates that increase in resistance to wear of composite with the increase in TiB2 percentage of reinforcement. According to the microstructural analysis, an increase in stirring speed and stirring time reduces porosity. X-ray diffraction was used to identify crystalline phases of various MMCs. A unit cell dimensions can also be calculated based on the diffraction peak positions obtained from the X-ray diffractometer (XRD) test. The process of determining the unit cell dimensions from peak positions is known as indexing. Miller indices, hkl, must be assigned to each peak in a powder diffraction pattern to index it. The process of indexing electron diffraction patterns of a single crystal is to label the individual diffraction spots with their proper values and sign (+or -) of h, k, and l. When x-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the condition, the Angle of incidence = Angle of scattering and the path length integer is equal to an integer number of wavelengths. The condition for maximum intensity contained in Bragg's law above allows us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the x-rays incident upon the crystal. Indexing of diffraction patterns was done for all the metal matrix composites and in which the miller indices and interplanar d- spacing were calculated.

A Pathfinding Problem for Fork-Join Directed Acyclic Graphs with Unknown Edge Length

In a previous paper by the author, a pathfinding problem for directed trees is studied under the following situation: each edge has a nonnegative integer length, but the length is unknown in advance and should be found by a procedure whose computational cost becomes exponentially larger as the length increases. In this paper, the same problem is studied for a more general class of graphs called fork-join directed acyclic graphs. The problem for the new class of graphs contains the previous one. In addition, the optimality criterion used in this paper is stronger than that in the previous paper and is more appropriate for real applications.

MacMahon Partition Analysis: A discrete approach to broken stick problems

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for n≥k≥3, the probability that a k-gon cannot be formed from a stick broken into n parts is given by n! over a product of linear combinations of partial sums of generalized Fibonacci numbers, a problem which proved to be very hard to generalize in the past.

THE DEVELOPMENT OF PEDAGOGICAL SCIENCE IN KARAKALPAKSTAN IN THE MIDDLE OF THE 20TH CENTURY - THE BEGINNING OF THE 21ST CENTURY

This article reviews and describes the analysis called on earlier studies on shaping and development of the pedagogical science in Karakalpakstan in 20th century in it is considered main questions and problems of the formation and sciences in the Republic of Karakalpakstan on length integer age. The review is given in chronological order. The presented name that scientist, who lifted, considered or tried to answer the questions in the field of theories and histories pedagogy in its epoch. The article gives short analysis leading period to pedagogical thought of this time. With highlights of the histories author made a stab corresponding to findings. The article deals with the rich informative material. The researchers concerning with questions of the formation and history of pedagogy might find their answers.

Efficient Private Comparison Queries Over Encrypted Databases Using Fully Homomorphic Encryption With Finite Fields

To achieve security and privacy for data stored on the cloud, we need the ability to secure data in compute. Equality comparisons, “ <inline-formula><tex-math notation="LaTeX">$x=y, x\ne y$</tex-math></inline-formula> ”, have been widely studied with many proposals but there is much room for improvement for order comparisons, “ <inline-formula><tex-math notation="LaTeX">$x &lt; y,~x \leq y,~x &gt; y \text{ and } x \geq y$</tex-math></inline-formula> ”. Most protocols for order comparisons have some limitation, either leaking some information about the data or requiring several rounds of communication between client and server. In addition, little work has been done on retrieving with compound conditions, mixing several equality and order comparisons. Fully homomorphic encryption (FHE) promises the ability to compute arbitrary functions on encrypted data without sacrificing privacy and without communication, but its potential has yet to be fulfilled. Particularly, private comparisons for database queries using FHE are expensive to compute. In this article, we design an efficient private database query (PDQ) protocol which supports compound conditions with equality and order comparisons. To this end, we first present a private comparison algorithm on encrypted integers using FHE, which scales efficiently for the length of input integers, by applying techniques from finite field theory. Then, we consider a scenario for PDQ protocols, querying for values based on a conjunction of one order and four equality conditions on key columns. The proposed algorithm and protocol are implemented and tested to determine their performance in practice. The proposed comparison algorithm takes about <inline-formula><tex-math notation="LaTeX">$25.259$</tex-math></inline-formula> seconds to compare 697 pairs of 64-bit integers using Brakerski-Gentry-Vaikuntanathan's leveled FHE scheme with single instruction multiple data (SIMD) techniques at more than 138 bits of security. This yields an amortized rate of just 36 milliseconds per comparison. On top of that, we show that our techniques achieve an efficient PDQ protocol for one order and four equality comparisons, achieving an amortized time and communication cost of 57 milliseconds and 448 bytes per database element.

Quad Layouts via Constrained T‐Mesh Quantization

AbstractWe present a robust and fast method for the creation of conforming quad layouts on surfaces. Our algorithm is based on the quantization of a T‐mesh, i.e. an assignment of integer lengths to the sides of a non‐conforming rectangular partition of the surface. This representation has the benefit of being able to encode an infinite number of layout connectivity options in a finite manner, which guarantees that a valid layout can always be found. We carefully construct the T‐mesh from a given seamless parametrization such that the algorithm can provide guarantees on the results' quality. In particular, the user can specify a bound on the angular deviation of layout edges from prescribed directions. We solve an integer linear program (ILP) to find a coarse quad layout adhering to that maximal deviation. Our algorithm is guaranteed to yield a conforming quad layout free of T‐junctions together with bounded angle distortion. Our results show that the presented method is fast, reliable, and achieves high quality layouts.

Lattice-based zero-knowledge arguments for additive and multiplicative relations

In this work, we propose new lattice-based protocols which are used to prove additive and multiplicative relations of committed integers. We introduce three new protocols. The first protocol proves additive relation of integers. In this framework, we introduce a new computational technique which splits the integers into chunks helping to achieve a significant improvement to the integer addition protocol proposed at CRYPTO’18 by reducing the computational costs significantly for commonly used integers of length $$L\in \{2^5,2^6,2^7\}$$ . Our second protocol presents a new way of proving multiplicative relations of polynomials and improves the performance of the existing polynomial multiplication protocol proposed at ESORICS’15 for small integers. Using these two developed protocols as building blocks, we present our third contribution to prove multiplicative relation of integers and achieve a notable reduction in computational complexity compared to the existing integer multiplication protocol presented at CRYPTO’18.

An Audio Steganography Based on Run Length Encoding and Integer Wavelet Transform

This paper proposes an audio steganography method based on run length encoding and integer wavelet transform which can be used to hide secret message in digital audio. The major contribution of the proposed scheme is to propose an audio steganography with high capacity, where the secret information is compressed by run length encoding. In the applicable scenario, the main purpose is to hide as more information as possible in the cover audio files. First, the secret information is chaotic scrambling, then the result of scrambling is run length encoded, and finally, the secret information is embedded into integer wavelet coefficients. The experimental results and comparison with existing technique show that by utilizing the lossless compression of run length encoding and anti-attack of wavelet domain, the proposed method has improved the capacity, good audio quality, and can achieve blind extraction while maintaining imperceptibility and strong robustness.

Long gaps in sieved sets

For each prime p , let I_p \subset \mathbb{Z}/p\mathbb{Z} denote a collection of residue classes modulo p such that the cardinalities |I_p| are bounded and about 1 on average. We show that for sufficiently large x , the sifted set \{n \in \mathbb{Z}: n (\mathrm {mod}\: p) \not \in I_p \: \mathrm {for \: all} p \leq x\} contains gaps of size x (\log x)^{\delta} depends only on the densitiy of primes for which I_p \neq \emptyset . This improves on the "trivial'' bound of \gg x . As a consequence, for any non-constant polynomial f: \mathbb{Z} \to \mathbb{Z} with positive leading coefficient, the set \{n \leq X: f(n) \; \mathrm {composite}\} contains an interval of consecutive integers of length \ge (\log X) (\log\log X)^{\delta} for sufficiently large X , where \delta &gt; 0 depends only on the degree of f .

An efficient algorithm for packing cuts and (2,3)-metrics in a planar graph with three holes

We consider a planar graph G in which the edges have nonnegative integer lengths such that the length of every cycle of G is even, and three faces are distinguished, called holes in G. It is known that there exists a packing of cuts and (2,3)-metrics with nonnegative integer weights in G which realizes the distances within each hole. We develop a purely combinatorial strongly polynomial-time algorithm to find such a packing.

On the Zhang–Zagier measure

We first improve the known lower bound for the absolute Zhang–Zagier measure in the general case. Then we restrict our study to totally positive algebraic integers. In this case, we are able to find six points for the related spectrum. At last, we give inequalities involving the Zhang–Zagier measure, the Mahler measure and the length of such integers.

Merit factor analysis of polyphaser sequences using cyclic algorithm new with good correlation properties

Polyphase sequences such as Pn (n=1, 2, 3, 4, x), Golomb, Frank, and the Chu are with good correlation properties, lower sidelobe levels and large merit factor values are helpful in applications like radar, sonar and channel estimation and communications. The goodness of a sequence obtained from merit factor. The transmitted and received signal may not be the same due to noise. The correlation function of given sequence is expressed by ISL (Integrated Sidelobe Level) by minimizing the ISL metrics the performance parameter merit factor is improved. To make this possible the ISL metric is expressed in the frequency domain and minimized to its most recent values and fixing at their most recent value until the predefined threshold satisfied. Because of FFT operations, the Cyclic Algorithm New applied to very long length sequences say N~106. In this paper, the Merit factor and correlation levels compared with standard, and cyclic algorithm new initialized with Polyphase sequences for lengths 102~104. Moreover, the observations made for four consecutive even and odd integer lengths say 162, 172, 182, and 192. CAN (P3, Golomb) exhibits merit factor improvement of 3.77%. These sequences of sidelobe levels reduced.

A pathfinding problem for search trees with unknown edge length

We study a pathfinding problem of directed trees under the following situation: each edge has a nonnegative integer length, but the length is unknown in advance and should be searched by a procedure whose computational cost becomes very large as the length increases. We present a method that gives a shortest path. Under some assumptions, the proposed method is optimal in the number of procedure calls for the longest edge length. This graph problem is motivated by procedure synthesis for system configuration in cloud systems.

Level Set Estimates for the Discrete Frequency Function

We introduce the discrete frequency function as a possible new approach to understanding the discrete Hardy–Littlewood maximal function. Considering that the discrete Hardy–Littlewood maximal function is given at each integer by the supremum of averages over intervals of integer length, we define the discrete frequency function at that integer as the value at which the supremum is attained. After verifying that the function is well-defined, we investigate size and smoothness properties of this function.

Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations

We consider elliptic differential operators on either the entire Euclidean space Rd or subsets consisting of a cube ΛL of integer length L. For eigenfunctions of the operator, and more general solutions of elliptic differential equations, we derive several quantitative unique continuation results. The first result is of local nature and estimates the vanishing order of a solution. The second is a sampling result and compares the L2-norm of a solution over a union of equidistributed δ-balls in space with the L2-norm on the entire space. In the case where the space Rd is replaced by a finite cube ΛL, we derive similar estimates. Particular features of our bound are that they are uniform as long as the coefficients of the operator are chosen from an appropriate ensemble, they are quantitative and explicit with respect to the radius δ, and they are L-independent and stable under small shifts of the δ-balls. Our proof applies to second order terms which have slowly varying coefficients on the relevant length scale. The results can also be interpreted as special cases of uncertainty relations, observability estimates, or spectral inequalities.

More Grade School Triangles

Planar triangles with each side having a positive integer length are among the simplest geometrical objects imaginable. What can be said about the angles of such triangles? In particular, are these angles rational or at least algebraic numbers when measured in degrees? This article demonstrates that the answer in general is negative, except for three distinguished families of triangles. One family is well known since antiquity: With the largest angle equal to 90 degrees, it simply is the family of Pythagorean right triangles. Though not nearly as well-known, the other two families also deserve to be part of every geometry teacher's toolkit.