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In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small. [ABSTRACT FROM AUTHOR]

In the current paper, we investigate the following new class of system of difference equations un+1 =f􀀀1 g(vn􀀀1) A1 f (un􀀀2)+B1g(vn􀀀4) C1 f (un􀀀2)+D1g(vn􀀀4) vn+1 =g􀀀1 f (un􀀀1) A2g(vn􀀀2)+B2 f (un􀀀4) C2g(vn􀀀2)+D2 f (un􀀀4) n 2 N0; where the initial conditions u􀀀p, v􀀀p, for p = 0;4 are real numbers, the parameters Ar, Br, Cr, Dr, for r 2 f1;2g are real numbers, A2r +B2r 6= 0 6= C2 r +D2r, for r 2 f1;2g, f and g are continuous and strictly monotone functions, f (R) = R, g(R) = R, f (0) = 0, g(0) = 0. In addition, we solve aforementioned general two dimensional system of difference equations of fifth-order in explicit form. Moreover, we obtain the solutions of mentioned system according to whether the parameters being zeros or not. Finally, we present an interesting application. [ABSTRACT FROM AUTHOR]

In the present paper, we study relative difference sets (RDSs) and linked systems of them. It is shown that a closed linked system of RDSs is always graded by a group. Based on this result, we also define a product of RDS linked systems sharing the same grading group. Further, we generalize the Davis-Polhill-Smith construction of a linked system of RDSs. Finally, we construct new linked system of RDSs in a Heisenberg group over a finite field and family of RDSs in an extraspecial p-group of exponent p 2 . All constructions of new RDSs and their linked systems make usage of cyclotomic Schur rings. [ABSTRACT FROM AUTHOR]

Signed difference sets have interesting applications in communications and coding theory. A (v , k , λ) -difference set in a finite group G of order v is a subset D of G with k distinct elements such that the expressions x y - 1 for all distinct two elements x , y ∈ D , represent each non-identity element in G exactly λ times. A (v , k , λ) -signed difference set is a generalization of a (v , k , λ) -difference set D, which satisfies all properties of D, but has a sign for each element in D. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes. [ABSTRACT FROM AUTHOR]

Suppose a (λ n , n , λ n , λ) relative difference set exists in an abelian group G = S × H , where | S | = λ , | H | = n 2 , gcd (λ , n) = 1 , and λ is self-conjugate modulo λ n . Then λ is a square, say λ = u 2 , and exp (S) divides u by Turyn's exponent bound. We classify all such relative difference sets with exp (S) = u . We also show that n must be a prime power if an abelian (λ n , n , λ n , λ) RDS with gcd (λ , n) = 1 exists and λ is self-conjugate modulo n. [ABSTRACT FROM AUTHOR]

Quantum error correction lies at the heart of building reliable quantum information processing systems. Stabilizer codes, a fundamental class of quantum errorcorrecting codes, play a pivotal role in mitigating the adverse effects of noise and decoherence in quantum systems. This paper introduces a novel construction of quantum stabilizer codes using Hadamard difference sets, an elegant mathematical concept derived from combinatorial design theory. In this paper, the construction of the quantum stabilizer codes over non- cyclic Hadamard difference sets with parameters (4m²,2m²-m, m²-m), where m is a positive integer is discussed. Firstly, the parity check matrices are constructed from the Circulant permutation matrices with the help of Hadamard difference sets and then, the Symplectic inner product condition for Hadamard difference sets over binary operation for parity check matrices are obtained to affirm the commutative condition for Stabilizer operators which is vital for the error detection. For application, we constructed a Hadamard difference sets with parameters (16,6,2) for m = 2 of ordered pair of the group Z2 Z8 × (non-cyclic group) and quantum stabilizer codes are obtained by parity-check matrix. [ABSTRACT FROM AUTHOR]

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research. [ABSTRACT FROM AUTHOR]

In cognitive radio networks (CRNs), rendezvous is the vital step prior to the communication between two unlicensed secondary users (SUs), where the SUs hop on the same channel at the same time to establish a link. With the dramatic fall in the cost and size of wireless transceivers, it becomes more reasonable to apply multiple radios to achieve significant improvement in the rendezvous performance. However, most of the existing multiradio rendezvous algorithms are proposed for homogeneous CRNs where all the SUs are equipped with an equal number of radios and do not possess backward compatibility to SU with a single radio. In reality, the CRNs are heterogeneous in nature as SUs may have different numbers of radios. In this paper, a composite CH algorithm is proposed for an asynchronous and heterogeneous network to achieve blind rendezvous with full rendezvous diversity. An SU with m number radios are categorized into three groups those follow different channel hopping (CH) algorithms. The upper bound of the rendezvous latency is being evaluated with a brief theoretical and mathematical analysis. Extensive simulations have conducted for different performance metrics, and the results are compared with the state-of-art algorithms. Overall, the proposed algorithm shows better performance in heterogeneous CRNs. [ABSTRACT FROM AUTHOR]

We give a construction of a convex set A⊂R$A \subset \mathbb {R}$ with cardinality n$n$ such that A−A$A-A$ contains a convex subset with cardinality Ω(n2)$\Omega (n^2)$. We also consider the following variant of this problem: given a convex set A$A$, what is the size of the largest matching M⊂A×A$M \subset A \times A$ such that the set {a−b:(a,b)∈M}$$\begin{equation*} \lbrace a-b: (a,b) \in M \rbrace \end{equation*}$$is convex? We prove that there always exists such an M$M$ with |M|⩾n$|M| \geqslant \sqrt n$, and that this lower bound is best possible, up to a multiplicative constant. [ABSTRACT FROM AUTHOR]

Let p be a prime and let r , s be positive integers. In this paper, we prove that the Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$ , $x,y,m,n \in {\mathbb {N}}$ , $\min \{x,y\}>1$ , $\min \{m,n\}>2$ with $(x,y)=(p^r,p^s+1)$ has only one solution $(x,y,m,n)=(2,5,5,3)$. This result is related to the existence of some partial difference sets in combinatorics. [ABSTRACT FROM AUTHOR]

Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields. We present the first noncyclotomic infinite families of DPDFs which are also EPDFs, in structures other than finite fields (in particular cyclic groups and nonabelian groups). As well as direct constructions, we present an approach to constructing DPDFs/EPDFs using relative difference sets (RDSs); as part of this, we demonstrate how the well‐known RDS result of Bose extends to a very natural construction for DPDFs and EPDFs. [ABSTRACT FROM AUTHOR]

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group's structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields. [ABSTRACT FROM AUTHOR]

We show that the Freiman–Ruzsa theorem, characterising finite sets with bounded doubling, leads to an alternative proof of a characterisation of Meyer sets, that is, relatively dense subsets of Euclidean spaces whose difference sets are uniformly discrete. [ABSTRACT FROM AUTHOR]

In a recent work, A. Fish [Proc. Amer. Math. Soc. 146 (2018), pp. 3449–3453] proved that if E_{1} and E_{2} are two subsets of \mathbb {Z} of positive upper Banach density, then there exists k\in \mathbb {Z} such that k\cdot \mathbb {Z}\subset \left (E_{1}-E_{1}\right)\cdot \left (E_{2}-E_{2}\right). In this article we will show that a similar result is true for the set of primes \mathbb {P} (which has density 0). We will prove that there exists k\in \mathbb {N} such that k\cdot \mathbb {N}\subset \left (\mathbb {P}-\mathbb {P}\right)\cdot \left (\mathbb {P}-\mathbb {P}\right), where \mathbb {P}-\mathbb {P}=\left \{ p-q:p>q\text { and }p,q\in \mathbb {P}\right \}. [ABSTRACT FROM AUTHOR]

We investigate pairs of permutations F,G $F,G$ of Fpn ${{\mathbb{F}}}_{{p}^{n}}$ such that F(x+a)−G(x) $F(x+a)-G(x)$ is a permutation for every a∈Fpn $a\in {{\mathbb{F}}}_{{p}^{n}}$. We show that, in that case, necessarily G(x)=℘(F(x)) $G(x)=\wp (F(x))$ for some complete mapping −℘ $-\wp $ of Fpn ${{\mathbb{F}}}_{{p}^{n}}$, and call the permutation F $F$ a perfect ℘ $\wp $ nonlinear (P℘ $\wp $N) function. If ℘(x)=cx $\wp (x)=cx$, then F $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on Fpn×Fpn ${{\mathbb{F}}}_{{p}^{n}}\times {{\mathbb{F}}}_{{p}^{n}}$ involving ℘ $\wp $, we obtain a quasigroup, and show that the graph of a P℘ $\wp $N function F $F$ is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for P℘ $\wp $N functions, respectively, for the difference sets in the corresponding quasigroup. [ABSTRACT FROM AUTHOR]

Suppose that A is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set A - A is "not too large", then there is a nonzero group element with at least as many as (2 + o (1)) | A | 2 / | A - A | representations as a difference of two elements of A; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient 2 is best possible. We also prove continuous and multidimensional versions of this result, and obtain similar results for sufficiently dense subsets of an arbitrary abelian group. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory). [ABSTRACT FROM AUTHOR]

We consider the finite alphabet phase retrieval problem: recovering a signal whose entries lie in a small alphabet of possible values from its Fourier magnitudes. This problem arises in the celebrated technology of X-ray crystallography to determine the atomic structure of biological molecules. Our main result states that for generic values of the alphabet, two signals have the same Fourier magnitudes if and only if several partitions have the same difference sets. Thus, the finite alphabet phase retrieval problem reduces to the combinatorial problem of determining a signal from those difference sets. Notably, this result holds true when one of the letters of the alphabet is zero, namely, for sparse signals with finite alphabet, which is the situation in X-ray crystallography. [ABSTRACT FROM AUTHOR]

A single‐change covering design (SCCD) is a v $v$‐set X $X$ and an ordered list ℒ ${\rm{ {\mathcal L} }}$ of b $b$ blocks of size k $k$ where every pair from X $X$ must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single‐change covering design, or more simply, a single‐change covering design. A single‐change covering design is circular when the first and last blocks also differ by one element. A single‐change covering design is minimum if no other smaller design can be constructed for a given v,k $v,k$. In this paper, we use a new recursive construction to solve the existence of circular SCCD(v,4,b $v,4,b$) for all v $v$ and three residue classes of circular SCCD(v,5,b $v,5,b$) modulo 16. We solve the existence of three residue classes of SCCD(v,5,b) $(v,5,b)$ modulo 16. We prove the existence of circular SCCD(2c(k−1)+1,k,c2(2k−2)+c) $(2c(k-1)+1,k,{c}^{2}(2k-2)+c)$, for all c≥1,k≥2 $c\ge 1,k\ge 2$, using difference methods. [ABSTRACT FROM AUTHOR]

The uniformity principle of temperature difference fields (TDFs) is applied in this study to improve the constructal optimization for "volume-point" heat conduction based on entransy dissipation rate (EDR) minimization without the premise of an optimal last-order construct, and the constructal optimization algorithm based on EDR minimization is simplified in this paper. The results further prove that the uniformity principle of TDF is consistent with entransy theory. The constructal optimization of "volume-point" heat conduction based on EDR minimization is conducted not only to lower the average temperature but also to obtain a more uniform TDF distribution. Through comparing the optimal results based on EDR minimization without the premise of an optimal last-order construct with those based on maximum temperature difference (MTD) minimization, some criteria and formulas for designing conductivity paths based on EDR minimization and MTD minimization are proposed, and the idea and method of improving constructal optimization via the variational principle are proposed. [ABSTRACT FROM AUTHOR]

Difference sets have wide applications in constructing sequences and codes in engineering and cryptography, and in imaging with coded masks in astronomical events and in medical events. In this paper, the relation between difference sets and binary sequences is presented and a coded mask as an application of difference sets is given. First of all, an algorithm for an appropriate difference set is written in C++ by using twin primes and then a coded mask for imaging astronomical events is designed with the aid of this algorithm. [ABSTRACT FROM AUTHOR]

An equi-isoclinic tight fusion frame (EITFF) is a type of Grassmannian code, being a sequence of subspaces of a finite-dimensional Hilbert space of a given dimension with the property that the smallest spectral distance between any pair of them is as large as possible. EITFFs arise in compressed sensing, yielding dictionaries with minimal block coherence. Their existence remains poorly characterized. Most known EITFFs have parameters that match those of one that arose from an equiangular tight frame (ETF) in a rudimentary, direct-sum-based way. In this paper, we construct new infinite families of non-"tensor-sized" EITFFs in a way that generalizes the one previously known infinite family of them as well as the celebrated equivalence between harmonic ETFs and difference sets for finite abelian groups. In particular, we construct EITFFs consisting of Q planes in C Q for each prime power Q ≥ 4 , of Q − 1 planes in C Q for each odd prime power Q , and of 11 three-dimensional subspaces in R 11. The key idea is that every harmonic EITFF—one that is the orbit of a single subspace under the action of a unitary representation of a finite abelian group—arises from a smaller tight fusion frame with a nicely behaved "Fourier transform." Our particular constructions of harmonic EITFFs exploit the properties of Gauss sums over finite fields. [ABSTRACT FROM AUTHOR]

U.S. presidential primary debates are influential but under-researched. Before 2015, all of these debates, both Democratic and Republican, had 10 candidates or fewer. The first Republican debate in 2015, however, abided 17 candidates. They were split into two segments, with the 10 best-polling candidates in the main (prime-time) segment and the others in an 'undercard' session. A comparable pattern applied for the next six Republican debates. Concern arose not only because many candidates were crowded into a session but also because the undercard candidates were seen as receiving inferior exposure. The Democratic presidential primary debates that started four years later encountered similar difficulty. An authorized policy caused their candidates in each of the first two debates to be limited to 20, randomly divided into two groups of 10 appearing on successive nights. For remedy, this paper examines innovative debate plans, for different numbers of candidates, that feature symmetry among all candidates and entail many short segments with relatively few candidates in each. We apply combinatorial designs—balanced incomplete block designs and regular pairwise balanced designs, which are analogous to the games Spot It Jr.! Animals and (full-fledged) Spot It!, respectively. [ABSTRACT FROM AUTHOR]

Few studies have investigated the retreat of the East Asian summer monsoon (EASM) in comparison to its seasonal advances. This study examined the climatological characteristics of the EASM retreat based on the Japanese 55-year Reanalysis data and multiple observation data. First, the retreat date of EASM is defined based on the reversal of 850 hPa meridional winds over coastal East Asia. Then, the climatological characteristics of the EASM retreat are investigated based on the retreat date of the EASM. At lower troposphere, the differences of the 850-hPa winds between after and before the EASM retreat display an anticyclone (cyclone) over the East Asian continent (the subtropical western Pacific), resulting in the strengthening of northerly winds over the coastal region of East Asia. Such northerly winds lead to decrease of moisture transport over East Asia and bring colder air from higher latitudes to the coastal East Asia, both favoring the retreat of the EASM. At middle troposphere, the WPSH moves eastward, and a mid-latitude trough appears over northeast Asia. In addition, the difference fields show descending (ascending) motion and cold (warm) advection in the rear (front) of the mid-latitude trough. At upper troposphere, a divergent center exhibits a southeastward movement, which shifts from the Philippine Islands to the western North Pacific. As such, a convergence center appears over the coastal East Asia in the difference map. The retreat of the EASM is also associated with the decreased rainfall, the demise of the wet season, and the broadscale cooling over East Asia. [ABSTRACT FROM AUTHOR]

A (v , k , λ) difference set in a group G is a subset { d 1 , d 2 , ... , d k } of G such that D = ∑ d i in the group ring Z [ G ] satisfies D D - 1 = n + λ G , where n = k - λ . If D = ∑ s i d i , where the s i ∈ { ± 1 } , satisfies the same equation, we will call it a signed difference set. This generalizes both difference sets (all s i = 1 ) and circulant weighing matrices (G cyclic and λ = 0 ). We will show that there are other cases of interest, and give some results on their existence. [ABSTRACT FROM AUTHOR]

We investigate the Sidon set problem in the modular case and its corresponding version in Ramsey theory. Specifically, we study the function ̅ F(n) that maximizes the size of a Sidon set in Z n , as well as the minimum n such that Z n admits no Sidon r-partition (a partition whose parts are all Sidon sets), denoted by ̅ SR(r), for a fixed positive integer r. We use known results and the pigeonhole principle to establish an upper bound of ̅ SR(r), which allows us to find the exact values of ̅ SR(r) for r ϵ {2, 3, 4, 7}. We also present a criterion for determining the non-existence of almost difference sets (ADS) in Z n. By exploiting such criterion and the connection between ADS sets and the existence of Sidon sets in Z n , we derive nontrivial upper bounds of ̅ F(n) for infinitely many values of n, and we refine the upper bound on ̅ SR(r) in multiple cases, determining also the exact value for r ϵ {5, 6}. Our findings shed new light on the behavior of Sidon sets in cyclic groups. In particular, we find infinitely many values for which ̅ F(n) > ̅ F(n + 1). [ABSTRACT FROM AUTHOR]

A linear code is called a locally repairable code (LRC) with locality r if one can recover an erased code symbol by accessing at most r other code symbols. Constructions of LRCs have been widely investigated in recent years. In this paper, we give a step forward in this direction. Firstly, we propose a novel concept of perfect LRCs whose size exactly achieves the Hamming-type bound, similar to the perfect codes that achieving the Hamming bound in classical coding theory. By the parity-check matrix approach, we establish some important connections between the existence of LRCs and the existence of some subsets of finite geometry and finite fields with certain properties, respectively. By employing q-Steiner systems and sunflowers in projective geometry and difference sets in finite fields, we obtain two new constructions of perfect LRCs with flexible parameters and present several new constructions of k-optimal LRCs achieving another Hamming-type bound under the integers restriction. Moreover, for fixed q and r, the code lengths of all the q-ary r-LRCs constructed in this paper can be arbitrarily large and the code rates can asymptotically achieve the upper bound r r + 1 . [ABSTRACT FROM AUTHOR]

We present difference-dedicated second-order vibrational perturbation theory (VPT2) as an efficient method for the computation of nuclear magnetic resonance (NMR) isotopic shifts, which reflect the geometry dependence of the NMR property in combination with different vibration patterns of two isotopologues. Conventional calculations of isotopic shifts, e.g., by standard VPT2, require scanning the geometry dependence over the whole molecule, which becomes expensive rapidly as the molecule size increases. In DD-VPT2, this scan can be restricted to a small region around the substitution site. At the heart of DD-VPT2 is a set of localized vibration modes common to the two isotopologues and designed such that the difference between the vibration patterns is caught by a small subset of them (usually fewer than 10). We tested the DD-VPT2 method for a series of molecules with increasing size and found that this method provides results with the same quality as VPT2 and in good agreement with the experiment, with computational savings up to 95% and less numerical instabilities. The method is easy to automatize and straightforward to generalize to other molecular properties. [ABSTRACT FROM AUTHOR]

Let |$(a_n)_{n \geq 1}$| be a sequence of distinct positive integers. In a recent paper, Rudnick established asymptotic upper bounds for the minimal gaps of |$\{a_n \alpha \mod 1, ~1 \leq n \leq N\}$| as |$N \to \infty $|⁠ , valid for Lebesgue-almost all |$\alpha $| and formulated in terms of the additive energy of |$\{a_1, \dots , a_N\}$|⁠. In the present paper, we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of |$\{a_1, \dots , a_N\}$|⁠. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap and prove (slightly weaker) general upper and lower bounds that allow for a direct application. A major input for these results comes from the recent proof of the Duffin–Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick, and Radziwiłł on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard. [ABSTRACT FROM AUTHOR]

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We characterize the existence of an exponential dichotomy for a difference equation with delay in terms of the invertibility of a certain linear operator between quite general admissible spaces. These include all ℓ p spaces with p ∈ [ 1 , + ∞ ] as well as many other Banach spaces. The characterization requires using special norms that involve the delay. [ABSTRACT FROM AUTHOR]

In this paper, we choose convolutional neural network (CNN) as the method to diagnosis weak fault of rolling bearings. In order to improve the training effect of CNN, different two-dimensional image conversion algorithms which include Gramian angular sum or difference fields, wavelet time-frequency diagram, Markov transition field are introduced in to convert one-dimensional time series of bearing vibration signals into images. To relieve the pressure of hardware calculation and shorten the time of training and validation, we use the piecewise aggregate approximation (PAA) to compress the data as much as possible while preserving the whole signal information. We add the batch normalization layers to avoid the gradient saturation problem of ReLU function and minibatch method is used to overcome the instability of stochastic gradient descent with momentum (SGDM) while designing CNN. Each kind of images are made as the training sample, and the results show that both the wavelet time-frequency diagram and the Gramian sum or difference angle field diagram can better identify the fault state, and the wavelet time-frequency diagram was relatively better. By comparing with different recurrent neural network (RNN) diagnosis models, the validity of the model was proved. At the same time, the model is applied to the performance degradation identification of fault parts, and the results show that the model can effectively identify the degradation of inner ring, outer ring and rolling body, while the accuracy of inner ring and the outer ring is better. This paper provides a new idea for weak fault diagnosis of rolling bearings. [ABSTRACT FROM AUTHOR]

Bent functions f : V n → F p play an important role in constructing partial difference sets, where V n denotes an n-dimensional vector space over F p , p is an odd prime. In [2-3], the so-called vectorial dual-bent functions are considered to construct partial difference sets. In [2], Çeşmelioğlu et al. showed that for certain vectorial dual-bent functions F : V n → V s , the preimage set of 0 for F forms a partial difference set. In [3], Çeşmelioğlu et al. showed that for a class of Maiorana-McFarland vectorial dual-bent functions F : V n → F p s , the preimage set of the squares (non-squares) in F p s ∗ for F forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for certain vectorial dual-bent functions F : V n → F p s , the preimage set of the squares (non-squares) in F p s ∗ for F and the preimage set of any coset of some subgroup of F p s ∗ for F form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non-)quadratic vectorial dual-bent functions. In this paper, we illustrate that many results of using weakly regular p-ary bent functions to construct partial difference sets are special cases of our results. In [2], the authors considered weakly regular p-ary bent functions f with f (0) = 0 . They showed that if such a function f is an l-form with g c d (l - 1 , p - 1) = 1 for some integer 1 ≤ l ≤ p - 1 , then f is vectorial dual-bent. We prove that the converse also holds, which answers one open problem proposed in [3]. [ABSTRACT FROM AUTHOR]

The difference set Δ s , t of two (nondeterministic, in general) transducers s , t is the set of all input words for which the output sets of the two transducers are not equal. When the two transducers realize homomorphisms, their difference set is the complement of the well known equality set of the two homomorphisms. However, we show that transducer difference sets result in Chomsky-like classes of languages that are different than the classes resulting from equality sets. We also consider the following word problem: given transducers s , t and input w , tell whether the output sets s (w) and t (w) are different. In general the problem is PSPACE -complete, but it becomes NP -complete when at least one of the given transducers has finite outputs. We also provide a PRAX (polynomial randomized approximation) algorithm for the word problem as well as for the NFA (in)equivalence problem. Our presentation of PRAX algorithms improves the original presentation. • The difference set of two transducers is the set of all input words for which the corresponding output sets are not equal. • We show that transducer difference sets result in language classes that are different than classes resulting from equality sets. • We also consider the following word problem: given transducers s , t and input w , tell whether s (w) and t (w) are different. • The problem is PSPACE-complete, but it becomes NP-complete when at least one of the given transducers has finite outputs. • We also provide polynomial randomized approximation (PRAX) algorithms for the word problem and for the NFA equivalence problem. • Our presentation of PRAX algorithms improves the original presentation. [ABSTRACT FROM AUTHOR]

Denniston constructed partial difference sets (PDSs) with the parameters (2 3 m , (2 m + r − 2 m + 2 r) (2 m − 1) , 2 m − 2 r + (2 m + r − 2 m + 2 r) (2 r − 2) , (2 m + r − 2 m + 2 r) (2 r − 1)) in elementary abelian groups of order 2 3 m for all m ≥ 2 , 1 ≤ r < m. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters (p 3 m , (p m + r − p m + p r) (p m − 1) , p m − p r + (p m + r − p m + p r) (p r − 2) , (p m + r − p m + p r) (p r − 1)) exist in all elementary abelian groups of order p 3 m for all m ≥ 2 , r ∈ { 1 , m − 1 } where p is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes. [ABSTRACT FROM AUTHOR]

We give explicit constructions of square (1 , − 1) -matrices of order congruent to 2 modulo 4 with large determinants. In particular, we construct nine families of near-Hadamard matrices based on almost supplementary difference sets with two blocks, and compute their determinants. As consequences, we obtain many (1 , − 1) -matrices which are almost D -optimal or have at least 80% D -efficiency. [ABSTRACT FROM AUTHOR]

This article proposes an analytical technique based on difference sets (DSs) and discrete fourier transform (DFT) for the synthesis of thinned planar antenna arrays. DSs are chosen as a baseline to prove and establish pattern synthesis rules on the peak sidelobe level (PSL), the directivity, and the first null beam width. DFT technique is employed for finding the nonuniform excitations of a planar antenna array. The hybrid DFT based DS method is designed in this article for the antenna array thinning with an objective of reduction in PSL. This method extracts the excitations by simple cyclic shifts from a suitable DS sequence. The results from a wide simulation are illustrated to validate the reliability and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Katznelson's Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson's Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form (x,t) \mapsto (x + \alpha, t + h(x)). We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers. [ABSTRACT FROM AUTHOR]

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain 'regular' covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets. [ABSTRACT FROM AUTHOR]

The recently introduced concept of a bent partition of a $2m$ -dimensional vector space $\mathbb {V}_{2m}^{(p)}$ over a prime field $\mathbb {F}_{p}$ exhibits similar properties as a partition from a spread. In particular, it gives rise to a large family of bent functions obtained in the same manner as spread bent functions. We show that the first non-spread construction of bent partitions introduced by Pirsic and the third author ($p=2$), respectively, the first and the third author ($p$ odd), gives rise to a large variety of different bent partitions. Especially, we show that the sets of bent functions obtained with any two such bent partitions do not intersect. We then show that every union of sets from one of these bent partitions always forms a partial difference set. This generalizes some known results on partial difference sets from spreads. Some general results on partial difference sets from bent partitions of $\mathbb {V}_{2m}^{(2)}$ are given in the last section. [ABSTRACT FROM AUTHOR]

For geophysical electromagnetic (EM) forward modeling problems, the accuracy of solutions mainly depends on the numerical modeling method used and the corresponding boundary conditions. Most multi-dimensional EM studies deal with numerical methods for discretisation (e.g., finite-difference, finite-element, integral equation, etc.) and pay less attention to the boundaries. This review presents the recent research on optimizing boundary conditions for the frequency-domain marine controlled-source EM (CSEM) forward modeling algorithm. Current geophysical EM field simulation techniques usually utilize the truncated Dirichlet boundary condition, which requires the modeling domain boundaries to be far away from the area of interest and field values to be zero at the boundaries to mitigate artificial reflections/refractions resulting from truncated boundaries. The perfectly matched layer (PML) approach with few additional absorbing layers can serve as an alternative boundary to supress these truncated boundary effects. In this review, the application of the PML boundary condition to marine CSEM using a staggered finite-difference scheme for the 2.5D problem in vertical transverse isotropic (VTI) conductivity structures is introduced. This new algorithm utilizes the complex frequency-shifted PML (CFS-PML) boundary condition. The selection of optimal PML parameters are also further investigated for numerical stability. Numerical tests for several Earth conductivity models show that the CFS-PML approach is of similar high accuracy compared to using traditional Dirichlet boundary condition and exhibits additional advantages in terms of computational time and memory usage. Furthermore, the numerical tests indicate that the proposed forward modeling algorithm using CFS-PML boundary condition works well for both shallow and deep water cases, including the application to real field example from the Troll Field in Norway. The detectability of subsurface-related EM fields in airwave dominated shallow waters can be enhanced by using the weighted difference fields for mitigating the effect of airwaves on the models. [ABSTRACT FROM AUTHOR]

The existence problem of difference sets in a group becomes more interesting since the applications of difference sets on real life problems become more common. There are several construction methods for difference sets: the relation among parameters, nonexistence of difference sets (Bruck Ryser Chowla Theorem), multipliers etc. A similar problem for symmetric designs along with an investigation of Bruck Ryser Chowla theorem has been discussed by the writers (Sakarya University Journal of Science). In this paper, we study the existence problem of difference sets in a more general concept by using difference sets parameters, BRC Theorem, and an algorithm written in MATLAB. [ABSTRACT FROM AUTHOR]

Synthesizing 7T Susceptibility Weighted Imaging (SWI) from 3T SWI could offer significant clinical benefits by combining the high sensitivity of 7T SWI for neurological disorders with the widespread availability of 3T SWI in diagnostic routines. Although methods exist for synthesizing 7T Magnetic Resonance Imaging (MRI), they primarily focus on traditional MRI modalities like T1-weighted imaging, rather than SWI. SWI poses unique challenges, including limited data availability and the invisibility of certain tissues in individual 3T SWI slices. To address these challenges, we propose a Self-supervised Anatomical Continuity Enhancement (SACE) network to synthesize 7T SWI from 3T SWI using plentiful 3T SWI data and limited 3T–7T paired data. The SACE employs two specifically designed pretext tasks to utilize low-level representations from abundant 3T SWI data for assisting 7T SWI synthesis in a downstream task with limited paired data. One pretext task emphasizes input-specific morphology by balancing the elimination of redundant patterns with the preservation of essential morphology, preventing the blurring of synthetic 7T SWI images. The other task improves the synthesis of tissues that are invisible in a single 3T SWI slice by aligning adjacent slices with the current slice and predicting their difference fields. The downstream task innovatively combines clinical knowledge with brain substructure diagrams to selectively enhance clinically relevant features. When evaluated on a dataset comprising 97 cases (5495 slices), the proposed method achieved a Peak Signal-to-Noise Ratio (PSNR) of 23.05 dB and a Structural Similarity Index (SSIM) of 0.688. Due to the absence of specific methods for 7T SWI, our method was compared with existing enhancement techniques for general 7T MRI synthesis, outperforming these techniques in the context of 7T SWI synthesis. Clinical evaluations have shown that our synthetic 7T SWI is clinically effective, demonstrating its potential as a clinical tool. [Display omitted] • The first self-supervised learning method is proposed for 7T SWI synthesis. • A novel morphology-aware inpainting pretext task to prevent synthetic image blurring. • An effective pretext task to utilize spatial-anatomical context in low-level vision. • A new loss function combining clinical knowledge to bridge the research-practice gap. [ABSTRACT FROM AUTHOR]

In radar sensing and communications, designing Doppler resilient sequences (DRSs) with low ambiguity function for delay over the entire signal duration and Doppler shift over the entire signal bandwidth is an extremely difficult task. However, in practice, the Doppler frequency range is normally much smaller than the bandwidth of the transmitted signal, and it is relatively easy to attain quasi-synchronization for delays far less than the entire signal duration. Motivated by this observation, we propose a new concept called low ambiguity zone (LAZ) which is a small area of the corresponding ambiguity function of interest defined by the certain Doppler frequency and delay. Such an LAZ will reduce to a zero ambiguity zone (ZAZ) if the maximum ambiguity values of interest are zero. In this paper, we derive a set of theoretical bounds on periodic LAZ/ZAZ of unimodular DRSs with and without spectral constraints, which include the existing bounds on periodic global ambiguity function as special cases. These bounds may be used as theoretical design guidelines to measure the optimality of sequences against Doppler effect. We then introduce four optimal constructions of DRSs with respect to the derived ambiguity lower bounds based on some algebraic tools such as characters over finite field and cyclic difference sets. [ABSTRACT FROM AUTHOR]

As a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Combin. Des. 27 (2019), 756–783]. In this paper, we continue the research in the paper mentioned above. We will obtain various characterizations of plateaued functions; these characterizations establish close connections between plateaued functions and some combinatorial objects: partial geometric difference sets and related partial geometric difference families. Then we introduce the complementary matrix and Cayley matrix for a subset of a finite group and use them to characterize partial geometric difference sets. As applications, we will show how to construct directed strongly regular graphs from partial geometric difference sets and establish a natural relation between partial geometric difference sets and partial geometric designs. The tensor product of a group algebra and a cyclotomic field is an important tool for our discussions. [ABSTRACT FROM AUTHOR]