Search

This work explores the extent to which LSB embedding can be made secure against structural steganalysis through a modification of cover image statistics prior to message embedding. Natural images possess symmetries that are expressed through approximately equal cardinalities of certain sets of $k$-tuples of consecutive pixels. LSB embedding disturbs this balance and a $k^{\rm th}$-order structural attack infers the presence of a hidden message with a length in proportion to the size of the imbalance amongst sets of $k$-tuples. To protect against $k^{\rm th}$-order structural attacks, cover modifications involve the redistribution of $k$-tuples among the different sets so that symmetries of the cover image are broken, then repaired through the act of LSB embedding so that the stego image bears the statistics of the original cover. To protect against all orders up to some order $k$, the statistics of $n$-tuples must be preserved where $n$ is the least common multiple of all orders $\leq k$. We find that this is only feasible for securing against up to $3^{\rm rd}$-order attacks (Sample Pairs and Triples analyses) since higher-order protections result in virtually zero embedding capacities. Securing up to $3^{\rm rd}$-order requires redistribution of sextuplets: rather than perform these $6^{\rm th}$-order cover modifications, which result in tiny embedding capacities, we reduce the problem to the redistribution of triplets in a manner that also preserves the statistics of pairs. This is done by embedding into only certain pixels of each sextuplet, constraining the maximum embedding rate to be $\leq 2/3$ bits per channel. Testing on a variety of image formats, we report best performance for JPEG-compressed images with a mean maximum embedding rate undetectable by $2^{\rm nd}$- and $3^{\rm rd}$-order attacks of 0.21 bits per channel., Comment: 23 pages, 6 figures. Section 3 added; revisions made to Section 6.3. Version accepted by Journal of Computer Security

Testing multiple hypotheses when observations are positively correlated is very common in practice. The dependence between observations can induce dependence between test statistics and distort the joint distribution of the true and false positives. It has a profound impact on the performance of common multiple testing procedures. While the marginal statistical properties of the true and false discoveries such as their means and variances have been extensively studied in the past, their correlation remains unknown.

Let G be a finite abelian group and let $$\exp (G)$$ denote the least common multiple of the orders of all elements of G. A $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrix is a G-invariant $$|G|\times |G|$$ matrix H whose entries are complex hth roots of unity such that $$HH^*=|G|I_{|G|}$$. By $$\nu _p(x)$$ we denote the p-adic valuation of the integer x. Using bilinear forms over abelian groups, we [11] constructed new classes of $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrices under the following conditions. The purpose of this paper is to further study the conditions on G and h so that a $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrix exists. We will focus on $${{\,\mathrm{\mathrm{BH}}\,}}({\mathbb {Z}}_n,h)$$ and $${{\,\mathrm{\mathrm{BH}}\,}}(G,2p^b)$$ matrices, where p is an odd prime. Our results describe various relation among |G|, $$\gcd (|G|,h)$$ and $${{\,\mathrm{\mathrm{lcm}}\,}}(|G|,h)$$. Moreover, they confirm the nonexistence of 623 cases in the 3310 open cases for the existence of $${{\,\mathrm{\mathrm{BH}}\,}}(\mathbb {Z}_n,h)$$ matrices in which $$1\le n,h\le 100$$. Finally, we show that $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrices can be used to construct a new family of perfect polyphase arrays.

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordell\`{e}s (2007), and extend certain related results of Hilberdink and T\'oth (2016). We also formulate some conjectures and open problems., Comment: 14 pages, revised

This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph G. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then G is called cop-win, and robber-win otherwise. In contrast to previous work, we study this classical combinatorial game on edge-periodic graphs. These are graphs with an infinite lifetime comprised of discrete time steps such that each edge e is assigned a bit pattern of length \(l_e\), with a 1 in the i-th position of the pattern indicating the presence of edge e in the i-th step of each consecutive block of \(l_e\) steps. Utilising the known framework of reachability games, we obtain an \(O(\textsf {LCM}(L)\cdot n^3)\) time algorithm to decide if a given n-vertex edge-periodic graph \(G^\tau \) is cop-win or robber-win as well as compute a strategy for the winning player (here, L is the set of all edge pattern lengths \(l_e\), and \(\textsf {LCM}(L)\) denotes the least common multiple of the set L). For the special case of edge-periodic cycles, we prove an upper bound of \(2\cdot l \cdot \textsf {LCM}(L)\) on the minimum length required of any edge-periodic cycle to ensure that it is robber-win, where \(l = 1\) if \(\textsf {LCM}(L) \ge 2\cdot \max L \), and \(l=2\) otherwise. Furthermore, we provide constructions of edge-periodic cycles that are cop-win and have length \(1.5 \cdot \textsf {LCM}(L)\) in the \(l=1\) case and length \(3\cdot \textsf {LCM}(L)\) in the \(l=2\) case.

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \begin{equation*} \log \operatorname{lcm} (F_1, F_2, \dots, F_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot n^2 \quad \text{as } n \to +\infty, \end{equation*} where $\operatorname{lcm}$ is the least common multiple and $\alpha := \big(1 + \sqrt{5}) / 2$ is the golden ratio. We prove that for every periodic sequence $\mathbf{s} = (s_n)_{n \geq 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_{\mathbf{s}} > 0$ such that \begin{equation*} \log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n) \sim \frac{3 \log \alpha}{\pi^2} \cdot C_\mathbf{s} \cdot n^2 , \quad \text{as } n \to +\infty . \end{equation*} Moreover, we show that if $(s_n)_{n \geq 1}$ is a sequence of independent uniformly distributed random variables in $\{-1,+1\}$ then \begin{equation*} \mathbb{E}\big[\log \operatorname{lcm} (F_3 + s_3, F_4 + s_4, \dots, F_n + s_n)\big] \sim \frac{3 \log \alpha}{\pi^2} \cdot \frac{15 \operatorname{Li}_2(1 / 16)}{2} \cdot n^2 , \quad \text{as } n \to +\infty , \end{equation*} where $\operatorname{Li}_2$ is the dilogarithm function.

For every positive integer n and every δ ∈ [ 0 , 1 ] , let B ( n , δ ) denote the probabilistic model in which a random set A ⊆ { 1 , … , n } is constructed by choosing independently every element of { 1 , … , n } with probability δ. Moreover, let ( u k ) k ≥ 0 be an integer sequence satisfying u k = a 1 u k − 1 + a 2 u k − 2 , for every integer k ≥ 2 , where u 0 = 0 , u 1 ≠ 0 , and a 1 , a 2 are fixed nonzero integers; and let α and β, with | α | ≥ | β | , be the two roots of the polynomial X 2 − a 1 X − a 2 . Also, assume that α / β is not a root of unity. We prove that, as δ n / log ⁡ n → + ∞ , for every A in B ( n , δ ) we have log ⁡ lcm ( u a : a ∈ A ) ∼ δ Li 2 ( 1 − δ ) 1 − δ ⋅ 3 log ⁡ | α / ( a 1 2 , a 2 ) | π 2 ⋅ n 2 with probability 1 − o ( 1 ) , where lcm denotes the lowest common multiple, Li 2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ = 1 . This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Matyas, who studied the deterministic case δ = 1 , and is motivated by an asymptotic formula for lcm ( A ) due to Cilleruelo, Rue, Sarka, and Zumalacarregui.

Greatest common factor (GCF) and least common multiple (LCM) have been two of the most difficult topics for primary students to grasp, especially when they are required to apply the concepts to solving real-life problems. Reported here is a quasi-experimental study of a comparison between a PBL approach and a traditional lecture method. The study recruited 55 Taiwanese sixth-grade students (ages 11-12), and examined the effects of PBL on their performance on GCF and LCM and on their attitudes towards mathematics. A comparison of the groups on pre and post-tests showed that the treatment group had a higher mean score on the post-test than the control group. Results of the students' responses to pre and post-questionnaires indicated that there were statistically significant differences between the groups on the post-questionnaire, implying that PBL has a positive impact on students' attitudes towards mathematics.

Purpose: To overcome the constraint of common multiple-baseline designs that only one case per stagger position is permitted.Methods: Three alternative strategies for assigning more than one case to each stagger position are examined.Results: The three recommended strategies achieve the objective while maintaining the study’s internal and statistical-conclusion validities.Conclusions: ExPRT, a freely available Excel-based randomization-test package, can be used to assist in both the design and statistical analysis associated with each of the strategies.

In this paper, we propose a novel general non-negative matrix factorization (general-NMF)-based digital watermarking scheme for copyright protection and integrity authentication of the image content. Specifically, the proposed general-NMF algorithm is able to factorize a matrix $C\in R_+^{s\times t}$ into a basis matrix $A\in R_+^{m\times n}$ and a coefficient matrix $B\in R_+^{p\times q}$ by removing the dimension-matching constraints required by the conventional NMF, where $s=m$ , $n = p$ , and $t= q$ . In particular, $s= m\cdot l/n$ , $t=l/p\cdot q$ , and the variable $l$ is the least common multiple of $n$ and $p$ . Furthermore, the generator factor of the random matrix and $n$ are used as the keys of the proposed digital watermarking scheme. Experimental results show that the proposed digital watermarking scheme can effectively resist various attacks and tampering.

For non-singular matrices with some restrictions, we establish the relationships between Smith normal forms and transforming matrices (a invertible matrices that transform the matrix to its Smith normal form) of two matrices with corresponding matrices of their least common right multiple over a commutative principal ideal domains. Thus, for such a class of matrices, given answer to the well-known task of M. Newman. Moreover, for such matrices, received a new method for finding their least common right multiple which is based on the search for its Smith normal form and transforming matrices.

The Kendrick mass defect (KMD) analysis of multiply charged polymeric distributions has recently revealed a surprising isotopic split in their KMD plots-namely a 1/z difference between KMDs of isotopes of an oligomer at charge state z. Relying on the KMD analysis of actual and simulated distributions of poly(ethylene oxide) (PEO), the isotopic split is mathematically accounted for and found to go with an isotopic misalignment in certain cases. It is demonstrated that the divisibility (resp. indivisibility) of the nominal mass of the repeating unit (R) by z is the condition for homolog ions to line up horizontally (resp. misaligned obliquely) in a KMD plot. Computing KMDs using a fractional base unit R/z eventually corrects the misalignments for the associated charge state while using the least common multiple of all the charge states as the divisor realigns all the points at once. The isotopic split itself can be removed by using either a new charge-dependent KMD plot compatible with any fractional base unit or the remainders of KM (RKM) recently developed for low-resolution data all found to be linked in a unified theory. These original applications of the fractional base units and the RKM plots are of importance theoretically to satisfy the basics of a mass defect analysis and practically for a correct data handling of single stage and tandem mass spectra of multiply charged homo- and copolymers. Graphical Abstract ᅟ.

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers $n$ such that $v_n$ is divisible by the least common multiple of $1, 2, \cdots, \lfloor {n^{1/4}}\rfloor $ has density one. In particular, for any positive integer $m$, the set of positive integers $n$ such that $v_n$ is divisible by $m$ has density one., Comment: 6 pages

So far, the concept of the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of two numbers are still being taught procedurally whether it is seeking multiples or by understanding prime numbers and prime factorization. This way often makes the students difficulties in determining the GCD and LCM. To overcome such things, the authors will offer a solution in learning mathematics material GCD and LCM in elementary school taught with Approach Realistic Mathematics Education (RME) food package presentation model. Mathematical learning with this approach is believed to make students learn as if they find themselves GCD and LCM concepts learned and will make learning more meaningful so that ultimately will improve students' understanding on GCD and LCM concepts.

Let { a i } i = 1 ∞ be a strictly increasing sequence of positive integers ( a i a j if i j ). In 1978, Borwein showed that for any positive integer n , we have ∑ i = 1 n 1 lcm ( a i , a i + 1 ) ≤ 1 − 1 2 n , with equality occurring if and only if a i = 2 i − 1 for 1 ≤ i ≤ n + 1 . Let 3 ≤ r ≤ 7 be an integer. In this paper, we investigate the sum ∑ i = 1 n 1 lcm ( a i , . . . , a i + r − 1 ) and show that ∑ i = 1 n 1 lcm ( a i , . . . , a i + r − 1 ) ≤ U r ( n ) for any positive integer n , where U r ( n ) is a constant depending on r and n . Further, for any integer n ≥ 2 , we also give a characterization of the sequence { a i } i = 1 ∞ such that the equality ∑ i = 1 n 1 lcm ( a i , . . . , a i + r − 1 ) = U r ( n ) holds.

This study aims to develop an RME-based hypothetical learning trajectory of least common multiple for elementary school students. This study used research & development based on the Plomp development model. The subjects of this study were 9fourth grade elementary school students in Pekanbaru, Riau. The data of this study was collected by distributing questionaires and conducting interviews. The data collected was analysed by using descriptive and quantitative analysis. The results indicated that the hypothetical learning trajectory developed was very valid with an average score of 84.37% and was very practical with an average score of 89.38% from the students and 88.89% from the teachers. Therefore, the RME-based hypothetical learning trajectory of least common multiple can be further used by elementary school students and teachers.

Let n and r be positive integers with r ≥ 2 and let A = { a i } i = 1 ∞ be a strictly increasing sequence of positive integers. Let S A , r ( n ) : = ∑ i = 1 n 1 lcm ( a i , … , a i + r − 1 ) . In 1978, Borwein showed that S A , 2 ( n ) ≤ 1 − 1 2 n with the equality occurring if and only if a i = 2 i − 1 for 1 ≤ i ≤ n + 1 . In 2017, Qian proved that S A , r ( n ) ≤ V r ( n ) for 3 ≤ r ≤ 7 and characterized the first n + r − 1 terms of the sequence A such that S A , r ( n ) = V r ( n ) holds, where V r ( n ) depends only on r and n . In this paper, we further investigate S A , r ( n ) for 8 ≤ r ≤ 11 and we obtain the least upper bound U r of S A , r ( n ) for all strictly increasing sequences A of positive integers and for all positive integers n , where U r is a constant depending only on r .

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t) is an element x in R, for which there exist polynomials q(t) and s(t) such that p(t)=q(t)(t-x)s(t). We investigate the rational relationships that hold among the pseudo-roots of p(t) by using the diamond operations for cover graphs of modular lattices., Minor corrections, to appear in "Journal of Pure and Applied Algebra"

Period estimation in one dimension (1-D) has been studied for years. However, 2-D period estimation is still a hard problem since it has three parameters to determine: length, width, and direction. Recently, a special kind of 2-D function called 2D-gcd-delta function is proposed. It has close relationship with Ramanujan's sum and the 2-D periodicity matrix. In this paper, we will describe how to use this function to decompose an image into subband signals. Each subband signal will have its own periodicity matrix so we also provide a simple algorithm to calculate the least common multiple (LCM) of those subband signal periodicity matrices. Concrete experiments are given to prove the robustness of the proposed 2-D period estimation method.

In this article, we establish a truncated non-integrated defect relation for meromorphic mappings from an m-dimensional complete Kahler manifold into P n ( C ) intersecting q hypersurfaces Q 1 , . . . , Q q in k-subgeneral position of degree d i , i.e., the intersection of any k + 1 hypersurfaces is emptyset. We will prove that ∑ i = 1 q δ f [ u − 1 ] ( Q i ) ≤ ( k − n + 1 ) ( n + 1 ) + ϵ + ρ u ( u − 1 ) d , where u is explicitly estimated and d is the least common multiple of d i ′ s. Our result generalizes and improves previous results. In the last part of this paper we will apply this result to study the distribution of the Gauss map of minimal surfaces.

The Fundamental Theorem of Arithmetic is a statement about the uniqueness of factorization in the ring of integers. The notation and proof easily generalize to uniqueness of factorization in principal ideal domains. Factorization, unique and otherwise, has been well-studied in integral domains. Less well-studied is factorization in rings that contain nonzero zerodivisors. In this article we give a theorem analogous to the Fundamental Theorem of Arithmetic, but for rings of the form D/(n) where D is a principal ideal domain and n is a nonzero nonunit of D.

As usual, writeIn [1], it is shown, by an elegant method, that for all n ⩾ 2, Hn is not an integer. The method can be traced back at least to [2, Exercise 251, p. 159]. Actually, rather more is shown: if Hn is expressed as a fraction an/bn in its lowest terms, then for all n ⩾ 2, bn is even. One way of looking at this is to say that once the term has entered the sum, the factor 2 persists in the denominator from then on. This suggests the following question:(Q1) for primes p ⩾ 3, is bn a multiple of p for all n ⩾ p?and more generally:(Q2) for any prime p, is bn a multiple of pm for all n ⩾ pm?In other words, once 1/pm has entered the sum, does the factor pm persist in the denominator?Let dn denote the lowest common multiple of 1, 2, … , n. This number can be described as follows: for each prime p ⩽ n, let mp be the largest integer m such that pm ⩽ n. Then . Hence a positive answer to (Q2) would imply that bn simply equals dn.

Let n be a positive integer and f ( x ) be a polynomial with nonnegative integer coefficients. We prove that lcm ⌈ n / 2 ⌉ ≤ i ≤ n { f ( i ) } ≥ 2 n − 1 ⌈ n / 2 ⌉ for any positive integer n, where ⌈ n / 2 ⌉ denotes the smallest integer that is not less than n / 2 . This improves the lower bound obtained by Hong, Luo, Qian and Wang in 2013. For the least common multiple of the first n positive integers, we show that lcm 1 ≤ i ≤ n { i } ≥ 2 n − 3 ( n − 1 ) n − 2 2 for any integer n ≥ 7 , which improves the lower bound obtained by Nair in 1982 and by Farhi in 2009. For the least common multiple of consecutive quadratic progression terms, by using the integration method combined with a little more detailed analysis on the absolute value of complex numbers, we further show that lcm ⌈ n / 2 ⌉ ≤ i ≤ n { a i 2 + c } ≥ 2 n − 1 ⌈ n / 2 ⌉ ⋅ min ⁡ ( a , a c ) for any positive integer n, where a and c are two positive integers. This improves and extends the results obtained by Farhi in 2005 and Oon in 2013, respectively.

Pairwise comparison is a very common multiple comparison problem. It is known that Fisher's LSD test does not control the familywise error rate (FWER) when there are more than three groups to be compared. Improved testing strategies include the Tukey-Kramer (TK) test that eliminates the F-test step and the two-step Fisher-Hayter (FH) test which requires a significant F-test. We propose a modified FH-test that is uniformly more powerful than the original version and relies on exact size α test under the balanced model. We provide simulations to show that the new procedure is preferred to the FH-test and the TK-test.

The problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli, namely the robust remaindering problem, may occur in many applications including phase unwrapping, frequency detection from several undersampled waveforms, wireless sensor networks, etc. Assuming that the dynamic range of the large number is the maximal possible one, i.e., the least common multiple (lcm) of all the moduli, a method called robust Chinese remainder theorem (CRT) for solving the robust remaindering problem has been recently proposed. In this paper, by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli, a trade-off between the dynamic range and the robustness bound for two-modular systems is studied. It basically says that a decrease in the dynamic range may lead to an increase of the robustness bound. We first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed-form formula for the robustness to hold. We then propose simple closed-form reconstruction algorithms. Furthermore, the newly obtained two-modular results are applied to the robust reconstruction for multi-modular systems and generalized to real numbers. Finally, some simulations are carried out to verify our proposed theoretical results., 32 pages, 5 figures

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $\varphi(n)^r$, $\sigma(n)^r$ ($r>-1$ real), where $\varphi$ is Euler's totient function and $\sigma$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function $f$, the function of $k$ variables $f([n_1,\ldots,n_k])$ is multiplicative., Comment: 12 pages

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ of degree $d\ge 2$ is irreducible over the rationals, then $\log\operatorname{lcm}(f(1),\ldots,f(N))\sim(d-1)N\log N$ as $N\to\infty$. He proved it for the case $d = 2$. Very recently, Maynard and Rudnick proved there exists $c_d > 0$ with $\log\operatorname{lcm}(f(1),\ldots,f(N))\gtrsim c_d N\log N$, and showed one can take $c_d = \frac{d-1}{d^2}$. We give an alternative proof of this result with the improved constant $c_d = 1$. We additionally prove the bound $\log\operatorname{rad}\operatorname{lcm}(f(1),\ldots,f(N))\gtrsim\frac{2}{d}N\log N$ and make the stronger conjecture that $\log\operatorname{rad}\operatorname{lcm}(f(1),\ldots,f(N))\sim (d-1)N\log N$ as $N\to\infty$.

This Note answers, and generalizes, a question of Kaisa Matom\"aki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $\chi$ such that $\pi_1\otimes\chi\simeq\pi_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $\pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$, Comment: The main result of the earlier version remains unchanged in this version. The main change is that the proof has been modified to "not use" anything about Galois representations or the local Langlands conjecture. Instead this new proof works totally on the "automorphic side", using only the representation theory of GL(n) and division algebras

Linguistic neutrosophic numbers (LNNs) are a powerful tool for describing fuzzy information with three independent linguistic variables (LVs), which express the degrees of truth, uncertainty, and falsity, respectively. However, existing LNNs cannot depict the hesitancy of the decision-maker (DM). To solve this issue, this paper first defines a hesitant linguistic neutrosophic number (HLNN), which consists of a few LNNs regarding an evaluated object due to DMs’ hesitancy to represent their hesitant and uncertain information in the decision-making process. Then, based on the least common multiple cardinality (LCMC), we present generalized distance and similarity measures of HLNNs, and then develop a similarity measure-based multiple-attribute decision-making (MADM) method to handle the MADM problem in the HLNN setting. Finally, the feasibility of the proposed approach is verified by an investment decision case.

A linguistic cubic hesitant variable (LCHV) is a hybrid form of linguistic values in group decision-making environments. It is composed of an interval language variable and multiple single-valued language variables given by different decision-makers (DMs). Due to the uncertainty and hesitation of DMs, the numbers of language variables in different LCHVs are unequal. Thus, the least common multiple number (LCMN) extension method was adopted. Based on the included angle and distance of two LCHVs, we presented two cosine similarity measures and developed a multiple attribute group decision-making (MAGDM) approach. An example of engineer selection was used to implement the proposed LCHV MAGDM method and demonstrate the simplicity and feasibility of the proposed method. The sensitivity analysis of weight changes for the two measures showed that the similarity measure based on distance was more stable than the similarity measure based on included angle in this application.

We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from $N$ non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any $T$ databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when $K>T/\textbf{gcd}(N,T)$ where $K$ is the total number of messages in the system and $\textbf{gcd}(\cdot,\cdot)$ means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as $\textbf{lcm}(N-T,T)$, where $\textbf{lcm}(\cdot,\cdot)$ means the least common multiple. On the other hand, when $K$ is small, we show that it is in fact possible to design codes with a message size even smaller than $\textbf{lcm}(N-T,T)$., 26 pages, 1 figure; accepted by IEEE Trans. Information Theory

In this work, the diffraction-based discrimination of two-dimensional (2D) orthogonal separable and nonseparable periodic structures and prediction of the reduced Talbot distances for 2D orthogonal nonseparable periodic structures are presented. 2D orthogonal periodic structures are defined and classified into separable (multiplicative or additive) and nonseparable categories with the aid of a spatial spectrum lattice. For both the separable and nonseparable cases, the spatial spectra or far-field impulses are 2D orthogonal lattices. We prove that for a 2D orthogonal separable structure, in addition to the DC impulse, there are other impulses on the coordinate axes. As a result, if all the spectrum impulses of a structure on the coordinate axes, except for the DC impulse, vanish, we conclude that the structure is nonseparable. In the second part of this work, using a unified formulation, the near-field diffraction of the 2D orthogonal separable and nonseparable periodic structures is investigated. In general, the Talbot distance equals the least common multiple of the individual Talbot distances in the orthogonal directions, say, z t =z lcm . For the 2D orthogonal nonseparable periodic structures having Fourier coefficients only with odd indices, we have found surprising results. It is shown that for this kind of structure, the Talbot distance strongly depends on the number theoretic properties of the structure. Depending on the ratio of the structure’s periods in the orthogonal directions, pxpy, the Talbot distance reduces to z lcm 2, z lcm 4, or z lcm 8. In addition, for the 2D orthogonal nonseparable sinusoidal grating, we show that, regardless of the value of pxpy, self-images are formed at distances smaller than the conventional Talbot distances attributed to px and py that we name the reduced Talbot (RT) distances. Halfway between two adjacent RT distances, the formation of negative self-images with a complementary amplitude of the self-images is predicted. Halfway between two adjacent self-image and negative-elf-image, subimages are formed. As another interesting result, we show that the intensity patterns of the subimages are 2D multiplicatively separable with halved periods in both directions. Finally, we show that 2D almost periodic structures with impulses on zone-plate-like concentric circles have self-images under plane wave illumination.

A novel algorithm for design concept generation using TRIZ inventive principles for resolving multi-parameter contradiction is described in the paper. The LCM algorithm inverts TRIZ contradiction matrix from parameter versus contradiction versus inventive principles form to solution versus parameter versus contradiction form followed by its bifurcation into two complementary matrices. The concept generation using the complementary matrices has been demonstrated for three case studies—the recursive evolution of inventive principles of algorithm by solving its own TRIZ contradiction matrix; high-performance diesel engine; and development of shrink fit and autofrettage concept for very high-pressure vessel. The first case study validates the algorithm. The recursive evolution leads an algorithm to give non-trivial solution due to positive definiteness as stated by theorem of recursive evolution. The algorithm leads to focussed heuristics and problem-oriented prioritisation of parameters by the bifurcation of solution set into basic inventive solution set and novel solution set. Scientific effect that can potentially constrain the applicability of contradiction resolution is reported. The work includes the updated contradiction matrix (Jou et al. in Adv Mater Sci Eng, 2013. https://doi.org/10.1155/2013/830891 ; Mann and Dewulf in TRIZ J, 2003).

Different from a general-purpose system, a real-time system requires stringent timing guarantees. While existing offline analysis techniques can provide timing guarantees using the worst-case execution time (WCET) of individual tasks, a variation of actual execution time makes it difficult to build covert timing channel. In this paper, we first present a novel covert timing channel, which considers actual execution time distribution of tasks and controls execution time to leak data between conspirators; we demonstrate that it is possible to leak data in real-time systems. Second, we suggest two enhancing techniques called S-R LCM (sender-receiver least common multiple) and noise area to reduce noise in communication. Through simulations, we demonstrate that our covert timing channel can serve trade-off between transmission speed and accuracy; that is, it shows average 50.2%, 54.6% and 51.3% accuracy for 100 test cases with thresholds 0, 1.4 and 2.8. Average 58.4% accuracy is accomplished with best threshold values for 100 test cases, and the maximum accuracy for a single test case is recorded 100.0%.