Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables \(S_i\) (\(=\pm \)1) and Z(2) synaptic connection (gauge) variables \(J_{ij}\) (\(=\pm \)1). Its energy consists of the Hopfield term \(-c_1S_iJ_{ij}S_j\), double Hopfield term \(-c_2 S_iJ_{ij}J_{jk} S_k\), and the reverberation (triple Hopfield) term \(-c_3 J_{ij}J_{jk}J_{ki}\) of synaptic self interactions. For the case \(c_2=0\), its phase diagram in the \(c_3-c_1\) plane has been studied both for the symmetric couplings \(J_{ij}=J_{ji}\) and asymmetric couplings (\(J_{ij}\) and \(J_{ji}\) are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing \(c_2\), and examine its effect. We find that the \(c_2\) term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of \(S_i\) and recalling them and measure the performance directly by overlaps of \(S_i\). We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections.     

Efficient parallelizable hashing using small non-compressing primitives

A well-established method of constructing hash functions is to base them on non-compressing primitives, such as one-way functions or permutations. In this work, we present \(S^r\), an \(rn\)-to-\(n\)-bit compression function (for \(r\ge 1\)) making \(2r-1\) calls to \(n\)-to-\(n\)-bit primitives (random functions or permutations). \(S^r\) compresses its inputs at a rate (the amount of message blocks per primitive call) up to almost 1/2, and it outperforms all existing schemes with respect to rate and/or the size of underlying primitives. For instance, instantiated with the \(1600\)-bit permutation of NIST’s SHA-3 hash function standard, it offers about \(800\)-bit security at a rate of almost 1/2, while SHA-3-512 itself achieves only \(512\)-bit security at a rate of about \(1/3\). We prove that \(S^r\) achieves asymptotically optimal collision security against semi-adaptive adversaries up to almost \(2^{n/2}\) queries and that it can be made preimage secure up to \(2^n\) queries using a simple tweak.     

Fast L1–L2 Minimization via a Proximal Operator

This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as \(L_1\) minimization do not work well. Recently, the difference of the \(L_1\) and \(L_2\) norms, denoted as \(L_1\)–\(L_2\), is shown to have superior performance over the classic \(L_1\) method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the \(L_1\)–\(L_2\) metric, and it makes some fast \(L_1\) solvers such as forward–backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for \(L_1\)–\(L_2\). We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of \(L_1\)–\(L_2\) based on a difference-of-convex approach in the numerical experiments.     

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.     

Recursive Computation of Logarithmic Derivatives,Ratios, and Products of Spheroidal Harmonics and Modified Bessel Functions and Applications

Spheroidal harmonics and modified Bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics (\(P_n^m(x)\), \(Q_n^m(x)\), \(n\ge m\ge 0\), \(x>1\)), the oblate spheroidal harmonics (\(P_n^m(ix)\), \(Q_n^m(ix)\), \(n\ge m\ge 0\), \(x>0\)), and the modified Bessel functions (\(I_n(x)\), \(K_n(x)\), \(n\ge 0\), \(x>0\)) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.     

A Hybridized Discontinuous Galerkin Method with Reduced Stabilization

In this paper, we propose a hybridized discontinuous Galerkin (HDG) method with reduced stabilization for the Poisson equation. The reduce stabilization proposed here enables us to use piecewise polynomials of degree \(k\) and \(k-1\) for the approximations of element and inter-element unknowns, respectively, unlike the standard HDG methods. We provide the error estimates in the energy and \(L^2\) norms under the chunkiness condition. In the case of \(k=1\), it can be shown that the proposed method is closely related to the Crouzeix–Raviart nonconforming finite element method. Numerical results are presented to verify the validity of the proposed method.     

Multilevel quantum Otto heat engines with identical particles

A quantum Otto heat engine is studied with multilevel identical particles trapped in one-dimensional box potential as working substance. The symmetrical wave function for Bosons and the anti-symmetrical wave function for Fermions are considered. In two-particle case, we focus on the ratios of \(W^i\) (\(i=B,F\)) to \(W_s\), where \(W^\mathrm{B}\) and \(W^\mathrm{F}\) are the work done by two Bosons and Fermions, respectively, and \(W_s\) is the work output of a single particle under the same conditions. Due to the symmetrical of the wave functions, the ratios are not equal to 2. Three different regimes, low-temperature regime, high-temperature regime, and intermediate-temperature regime, are analyzed, and the effects of energy level number and the differences between the two baths are calculated. In the multiparticle case, we calculate the ratios of \(W^i_M/M\) to \(W_s\), where \(W^i_M/M\) can be seen as the average work done by a single particle in multiparticle heat engine. For other working substances whose energy spectrum has the form of \(E_n\sim n^2\), the results are similar. For the case \(E_n\sim n\), two different conclusions are obtained.     

Quantum codes from cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$

In this paper, we study quantum codes over \(F_q\) from cyclic codes over \(F_q+uF_q+vF_q+uvF_q,\) where \(u^2=u,~v^2=v,~uv=vu,~q=p^m\), and p is an odd prime. We give the structure of cyclic codes over \(F_q+uF_q+vF_q+uvF_q\) and obtain self-orthogonal codes over \(F_q\) as Gray images of linear and cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). In particular, we decompose a cyclic code over \(F_q+uF_q+vF_q+uvF_q\) into four cyclic codes over \(F_q\) to determine the parameters of the corresponding quantum code.     

Multi-party quantum private comparison protocol base on $$$$-imensional entangle states

In this paper, a novel quantum private comparison protocol with \(l\)-party and \(d\)-dimensional entangled states is proposed. In the protocol, \(l\) participants can sort their secret inputs in size, with the help of a semi-honest third party. However, if every participant wants to know the relation of size among the \(l\) secret inputs, these two-participant protocols have to be executed repeatedly \(\frac{l(l-1)}{2}\) times. Consequently, the proposed protocol needs to be executed one time. Without performing unitary operation on particles, it only need to prepare the initial entanglement states and only need to measure single particles. It is shown that the participants will not leak their private information by security analysis.     

A Bayesian approach to forecasting daily air-pollutant levels

Forecasting air-pollutant levels is an important issue, due to their adverse effects on public health, and often a legislative necessity. The advantage of Bayesian methods is their ability to provide density predictions which can easily be transformed into ordinal or binary predictions given a set of thresholds. We develop a Bayesian approach to forecasting PM\(_{10}\) and O\(_3\) levels that efficiently deals with extensive amounts of input parameters, and test whether it outperforms classical models and experts. The new approach is used to fit models for PM\(_{10}\) and O\(_3\) level forecasting that can be used in daily practice. We also introduce a novel approach for comparing models to experts based on estimated cost matrices. The results for diverse air quality monitoring sites across Slovenia show that Bayesian models outperform classical models in both PM\(_{10}\) and O\(_3\) predictions. The proposed models perform better than experts in PM\(_{10}\) and are on par with experts in O\(_3\) predictions—where experts already base their predictions on predictions from a statistical model. A Bayesian approach—especially using Gaussian processes—offers several advantages: superior performance, robustness to overfitting, more information, and the ability to efficiently adapt to different cost matrices.     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

No nexiste nce of n-qubit u nexte ndible product bases of size $$2^ n-5$$2 n-5

It is known that the n-qubit system has no unextendible product bases (UPBs) of cardinality \(2^n-1\), \(2^n-2\) and \(2^n-3\). On the other hand, the n-qubit UPBs of cardinality \(2^n-4\) exist for all \(n\ge 3\). We prove that they do not exist for cardinality \(2^n-5\).     

The hierarchical Petersen network: a new interconnection network with fixed degree

Network cost and fixed-degree characteristic for the graph are important factors to evaluate interconnection networks. In this paper, we propose hierarchical Petersen network (HPN) that is constructed in recursive and hierarchical structure based on a Petersen graph as a basic module. The degree of HPN(n) is 5, and HPN(n) has \(10^n\) nodes and \(2.5 \times 10^n\) edges. And we analyze its basic topological properties, routing algorithm, diameter, spanning tree, broadcasting algorithm and embedding. From the analysis, we prove that the diameter and network cost of HPN(n) are \(3\log _{10}N-1\) and \(15 \log _{10}N-1\), respectively, and it contains a spanning tree with the degree of 4. In addition, we propose link-disjoint one-to-all broadcasting algorithm and show that HPN(n) can be embedded into FP\(_k\) with expansion 1, dilation 2k and congestion 4. For most of the fixed-degree networks proposed, network cost and diameter require \(O(\sqrt{N})\) and the degree of the graph requires O(N). However, HPN(n) requires O(1) for the degree and \(O(\log _{10}N)\) for both diameter and network cost. As a result, the suggested interconnection network in this paper is superior to current fixed-degree and hierarchical networks in terms of network cost, diameter and the degree of the graph.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

The panpositionable panconnectedness of crossed cubes

In many parallel and distributed multiprocessor systems, the processors are connected based on different types of interconnection networks. The topological structure of an interconnection network is typically modeled as a graph. Among the many kinds of network topologies, the crossed cube is one of the most popular. In this paper, we investigate the panpositionable panconnectedness problem with respect to the crossed cube. A graph G is r-panpositionably panconnected if for any three distinct vertices x, y, z of G and for any integer \(l_1\) satisfying \(r \le l_1 \le |V(G)| - r - 1\), there exists a path \(P = [x, P_1, y, P_2, z]\) in G such that (i) \(P_1\) joins x and y with \(l(P_1) = l_1\) and (ii) \(P_2\) joins y and z with \(l(P_2) = l_2\) for any integer \(l_2\) satisfying \(r \le l_2 \le |V(G)| - l_1 - 1\), where |V(G)| is the total number of vertices in G and \(l(P_1)\) (respectively, \(l(P_2)\)) is the length of path \(P_1\) (respectively, \(P_2\)). By mathematical induction, we demonstrate that the n-dimensional crossed cube \(CQ_n\) is n-panpositionably panconnected. This result indicates that the path embedding of joining x and z with a mediate vertex y in \(CQ_n\) is extremely flexible. Moreover, applying our result, crossed cube problems such as panpositionable pancyclicity, panpositionably Hamiltonian connectedness, and panpositionable Hamiltonicity can be solved.     

Good and asymptotically good quantum codes derived from algebraic geometry

In this paper, we construct several new families of quantum codes with good parameters. These new quantum codes are derived from (classical) t-point (\(t\ge 1\)) algebraic geometry (AG) codes by applying the Calderbank–Shor–Steane (CSS) construction. More precisely, we construct two classical AG codes \(C_1\) and \(C_2\) such that \(C_1\subset C_2\), applying after the well-known CSS construction to \(C_1\) and \(C_2\). Many of these new codes have large minimum distances when compared with their code lengths as well as they also have small Singleton defects. As an example, we construct a family \({[[46, 2(t_2 - t_1), d]]}_{25}\) of quantum codes, where \(t_1 , t_2\) are positive integers such that \(1<t_1< t_2 < 23\) and \(d\ge \min \{ 46 - 2t_2 , 2t_1 - 2 \}\), of length \(n=46\), with minimum distance in the range \(2\le d\le 20\), having Singleton defect at most four. Additionally, by applying the CSS construction to sequences of t-point (classical) AG codes constructed in this paper, we generate sequences of asymptotically good quantum codes.     

A multiple reversible watermarking technique for fingerprint authentication

The classical fingerprint recognition system can be compromised at the database and the sensor. Therefore, to beef up security of the fingerprint recognition system, reversible watermarking is proposed to protect these two points. Reversible watermarks thwart manipulations, \({viz}\)., copy and replay attacks and ensure that native fingerprint recognition accuracy remains unaffected. Fingerprint-dependent watermark \(W_1\) authenticates the database and shields it against the copy attack. The second watermark \(W_2\) verifies fingerprint captured by the sensor and foil replay attack. \(W_2\) is derived from the higher order moments of the fingerprint. Divergence in the computed and extracted watermarks indicates loss of authenticity. The experimental results validate proposed hypothesis.     

Throughput maximization for speed scaling with agreeable deadlines

We study the following energy-efficient scheduling problem. We are given a set of n jobs which have to be scheduled by a single processor whose speed can be varied dynamically. Each job \(J_j\) is characterized by a processing requirement (work) \(p_j\), a release date \(r_j\), and a deadline \(d_j\). We are also given a budget of energy E which must not be exceeded and our objective is to maximize the throughput (i.e., the number of jobs which are completed on time). We show that the problem can be solved optimally via dynamic programming in \(O(n^4 \log n \log P)\) time when all jobs have the same release date, where P is the sum of the processing requirements of the jobs. For the more general case with agreeable deadlines where the jobs can be ordered so that, for every \(i < j\), it holds that \(r_i \le r_j\) and \(d_i \le d_j\), we propose an optimal dynamic programming algorithm which runs in \(O(n^6 \log n \log P)\) time. In addition, we consider the weighted case where every job \(J_j\) is also associated with a weight \(w_j\) and we are interested in maximizing the weighted throughput (i.e., the total weight of the jobs which are completed on time). For this case, we show that the problem becomes \(\mathcal{NP}\)-hard in the ordinary sense even when all jobs have the same release date and we propose a pseudo-polynomial time algorithm for agreeable instances.     

A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

A new weak Galerkin (WG) finite element method is developed and analyzed for solving second order elliptic problems with low regularity solutions in the Sobolev space \(W^{2,p}(\Omega )\) with \(p\in (1,2)\). A WG stabilizer was introduced by Wang and Ye (Math Comput 83:2101–2126, 2014) for a simpler variational formulation, and it has been commonly used since then in the WG literature. In this work, for the purpose of dealing with low regularity solutions, we propose to generalize the stabilizer of Wang and Ye by introducing a positive relaxation index to the mesh size h. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along the interior element edges. When the norm index \(p\in (1,2]\), we strictly derive that the WG error in energy norm has an optimal convergence order \(O(h^{l+1-\frac{1}{p}-\frac{p}{4}})\) by taking the relaxed factor \(\beta =1+\frac{2}{p}-\frac{p}{2}\), and it also has an optimal convergence order \(O(h^{l+2-\frac{2}{p}})\) in \(L^2\) norm when the solution \(u\in W^{l+1,p}\) with \(p\in [1,1+\frac{2}{p}-\frac{p}{2}]\) and \(l\ge 1\). It is recovered for \(p=2\) that with the choice of \(\beta =1\), error estimates in the energy and \(L^2\) norms are optimal for the source term in the sobolev space \(L^2\). Weak variational forms of the WG method give rise to desirable flexibility in enforcing boundary conditions and can be easily implemented without requiring a sufficiently large penalty factor as in the usual discontinuous Galerkin methods. In addition, numerical results illustrate that the proposed WG method with an over-relaxed factor \(\beta (\ge 1)\) converges at optimal algebraic rates for several low regularity elliptic problems.     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.