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The present paper, is an extension and development of a serious of research papers dealing with the semi-hollow lifting. Initially we proved semi hollow-lifting modules, by assuming H Module is a module with unique element satisfying (semi) S Ring over, and if H Module 1 and H Module 2 are immediate summoned from H Module such that H Module H M o d u l e 1 ∩ H M o d u l e 2 is hollow, then H M o d u l e 1 ∩ H M o d u l e 2 is an immediate summoned of H Module , We are Assume that be limited subsection of the total setZ, ago g = 〖⨁¬¬〗_(z ∈ Z) H_k be local immediate summoned →〖⨁¬¬〗_(u ∈ F) Hu is an immediate summand of H. Now, assume that x ∈ S then x∈〖⨁¬¬〗_ (u ∈ T) Hu. One can easily to display which∈〖⨁¬¬〗_ (u ∈ F) Hu, thus Rx⊆〖⨁¬¬〗_(u ∈ F) Hu wherever F is a subset of index set T. Since Rx≪_S H and 〖⨁¬¬〗_ (u ∈ F) H_t is an immediate summand of, Rx≪_S 〖⨁¬¬〗_(u ∈ F) H_u and henceRx≪_S f.

Obrađena je tema izračunljivo prebrojivih skupova. Izvedena je njihov konstrukcija preko definicije Turingovog stroja i izračunljive funkcije koristeći pritom Gödelovo numeriranje. Također, preko iste konstrukije su definirani česti matematički pojmovi kao što su izomorfizmi i permutacije, ali u kontekstu izračunljivosti. Potom su spomenute mnoge karakterizacije izračunljivo prebrojivih skupova kao i svojstva koja vežemo za njih. Jedna od najvažnijih karakterizacija je Teorem o nabrajanju (eng. Listing Theorem). Uveden je Teorem o rekurziji koji je temeljito obrađen u ovom radu. Za kraj je uveden pojam izračunljivo prebrojivih indeksa kao i svojstva koja vežemo za njih., The aim of this thesis is to explain computable enumerable sets and their properties. Their construction was derived by defining a Turing machine and a computable function using Gödel numbering. Common mathematical concepts such as isomorphisms and permutations are also defined using the same construction, but in the context of computability. Subsequently, many characterizations of computable sets were mentioned, as well as the properties we associate with them. One of the most important characterizations is the Listing Theorem. The Recursion Theorem is introduced and discussed in detail in this thesis. Finally, the concept of computable countable indices is introduced, as well as the properties we associate with connecting them.

Traditionally, ensembles of Slepian-Wolf (SW) codes are defined such that every bin of each $n$-vector of each source is randomly drawn under the uniform distribution across the sets $\{0,1,\ldots,2^{nR_X}-1\}$ and $\{0,1,\ldots,2^{nR_Y}-1\}$, where $R_X$ and $R_Y$ are the coding rates of the two sources, $X$ and $Y$, respectively. In a few more recent works, where only one source, say, $X$, is compressed and the other one, $Y$, serves as side information available at the decoder, the scope is extended to variable-rate S-W (VRSW) codes, where the rate is allowed to depend on the type class of the source string, but still, the random-binning distribution is assumed uniform within the corresponding, type-dependent, bin index set. In this expository work, we investigate the role of the uniformity of the random binning distribution from the perspective of the trade-off between the reliability (defined in terms of the error exponent) and the compression performance (measured from the viewpoint of the source coding exponent). To this end, we study a much wider class of random-binning distributions, which includes the ensemble of VRSW codes as a special case, but it also goes considerably beyond. We first show that, with the exception of some pathological cases, the smaller ensemble, of VRSW codes, is as good as the larger ensemble in terms the trade-off between the error exponent and the source coding exponent. Notwithstanding this finding, the wider class of ensembles proposed is motivated in two ways. The first is that it outperforms VRSW codes in the above-mentioned pathological cases, and the second is that it allows robustness: in the event of a system failure that causes unavailability of the compressed bit-stream from one of the sources, it still allows reconstruction of the other source within some controllable distortion.

We consider a specific version of the authors’ approach to the synthesis of bases of discrete orthogonal transforms (DOTs). The approach takes into account the relation between the structure of basis functions of a transform and the existence of a certain numeral system on the (multidimensional) index set of an input signal. In contrast to Chernov’s prototype paper “Discrete orthogonal transforms with bases generated by self-similar sequences” (2018), which was concerned with DOTs associated with irredundant numeral systems (where each index of the input signal has a unique representation in a chosen numeral system), in the present paper we study the case of the so-called complete numeral systems. In this case, there is no bijection between the set of input indices of DOTs and the set of their digital representations. Potentially, such statements of applied problems naturally appear in image recognition, artificial intelligence, theory of formal languages, mathematical programming, and other areas where the analyzed objects are characterized by many heterogeneous attributes, which can be quantitative, qualitative, and mixed. There may be several copies of each object, and the copies may have inconsistent descriptions, which must be considered and analyzed as a whole. Such objects with many attributes can be represented as multisets (“sets with repetitions”). Since discrete spectral analysis is a basic tool for solving the described problems in the classical “multiple” interpretation of objects, we try to extend some ideas and methods of spectral analysis to the case of multiset objects.

In this paper, we propose a method for determining all minimal representations of a face of a polyhedron defined by a system of linear inequalities. Main difficulties for determining prime and minimal representations of a face are that the deletion of one redundant constraint can change the redundancy of other constraints and the number of descriptor index pairs for the face can be huge. To reduce computational efforts in finding all minimal representations of a face, we prove and use properties that deleting strongly redundant constraints does not change the redundancy of other constraints and all minimal representations of a face can be found only in the set of all prime representations of the face corresponding to the maximal descriptor index set for it. The proposed method is based on a top-down search strategy, is easy to implement, and has many computational advantages. Based on minimal representations of a face, a reduction of degeneracy degrees of the face and ideas to improve some known methods for finding all maximal efficient faces in multiple objective linear programming are presented. Numerical examples are given to illustrate the method.

With the increasing integration of various renewable energy resources into electric power systems, the stochastic power flow calculation, which falls into the uncertainty propagation (UP) problems, has attracted renewed interest recently. The adaptive sparse pseudospectral approximation method (A-SPAM) is highly efficient to solve UP problems. However, there still exist two major drawbacks: 1) its global error estimate is not accurate enough to evaluate the actual approximation error; and 2) it is intended for single-output systems and there is no efficient method for its application in multi-output systems. In order to overcome these two defects, in this study a new adaptive sparse pseudospectral approximation method (NA-SPAM) is proposed based on two improvements on A-SPAM. First, a new global error estimate is presented which features much higher estimation accuracy. Second, a revised version of error estimates is given based on unification of the index set, which apparently improves the calculation efficiency for multi-output problems. At last, the effectiveness of the two improvements and NA-SPAM is validated by mathematical and practical cases.

For solving large scale linear systems, a fast convergent randomized Kaczmarz-type method was constructed in Bai and Wu (2018) [4] . In this paper, we propose a geometric probability randomized Kaczmarz (GPRK) method by introducing a new index set J k and three supervised probability criteria defined on J k from a geometric point of view. Linear convergence of GPRK is proved, and the way of argument for the analysis of GPRK also leads to new sharper upper bounds for the randomized Kaczmarz (RK) method and the greedy randomized Kaczmarz (GRK) method. In practice, GPRK is implemented with a simple geometric probability criterion, i.e., the most efficient one of the aforementioned three supervised probability criteria defined on J k . The numerical results demonstrate that GPRK is robust and efficient, and it is faster than GRK in most of the tests in the sense of computing time.

The categoricity spectrum of a computable structure S is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable presentations of S. The degree of categoricity of S is the least degree in the categoricity spectrum of S. This paper is a survey of results on categoricity spectra and degrees of categoricity for computable structures. We focus on the results about degrees of categoricity for linear orders and Boolean algebras. We construct a new series of examples of degrees of categoricity for linear orders.

New general upper and lower bounds for the Perron root of a nonnegative matrix, which involve nonempty proper subsets of the index set and the matrix sparsity pattern, are suggested, and some special cases are considered. Also the nonsingularity criteria related to the upper bounds presented, which generalize some known results on subclasses of nonsingular ℋ-matrices, are derived.

We find an exact estimate for the algorithmic complexity of the class of autostable ordered Abelian groups.

This study, based on the characteristics of the Chinese government venture capital (GVC), developed an index system that basically allows to identify potential on key aspects on a given start‐up, proposing a decision‐making method based on analytic hierarchy process (AHP) and intuitionistic fuzzy set–technique for order of preference by similarity to ideal solution (IFS‐TOPSIS) to evaluate and select Chinese GVC investment projects. A numerical example of how the proposed model was applied to a GVC investment project selection in China is provided. Our results showed that the adoption of this methodology can contribute to increase fairness and transparency in the evaluation process on GVCs, and the proposed index set is ideal for GVCs for start‐up selection.

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$. It is conjectured in \cite{KNO} that for each Dynkin node $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for the Langland dual ${\mathfrak g} ^L$. In this paper we show that at the spin node $k=6$, the family of perfect crystals given in \cite{KMN2} form a coherent family and show that its limit $B^{6,\infty}$ is isomorphic to the ultra-discretization of the positive geometric crystal we constructed in \cite{MP} for the affine Lie algebra $D_6^{(1)}$ which proves the conjecture in this case.

This paper presents new theoretical results on the multistability of switched recurrent neural networks with radial basis functions and state-dependent switching. By partitioning state space, applying Brouwer fixed-point theorem and constructing a Lyapunov function, the number of the equilibria and their locations are estimated and their stability/instability are analyzed under some reasonable assumptions on the decomposition of index set and switching threshold. It is shown that the switching threshold plays an important role in increasing the number of stable equilibria and different multistability results can be obtained under different ranges of switching threshold. The results suggest that switched recurrent neural networks would be superior to conventional ones in terms of increased storage capacity when used as associative memories. Two examples are discussed in detail to substantiate the effectiveness of the theoretical analysis.

In recent years, the cost index predictions of construction engineering projects are becoming important research topics in the field of construction management. Previous methods have limitations in reasonably reflecting the timeliness of engineering cost indexes. The recurrent neural network (RNN) belongs to a time series network, and the purpose of timeliness transfer calculation is achieved through the weight sharing of time steps. The long-term and short-term memory neural network (LSTM NN) solves the RNN limitations of the gradient vanishing and the inability to address long-term dependence under the premise of having the above advantages. The present study proposed a new framework based on LSTM, so as to explore the applicability and optimization mechanism of the algorithm in the field of cost indexes prediction. A survey was conducted in Shenzhen, China, where a total of 143 data samples were collected based on the index set for the corresponding time interval from May 2007 to March 2019. A prediction framework based on the LSTM model, which was trained by using these collected data, was established for the purpose of cost index predictions and test. The testing results showed that the proposed LSTM framework had obvious advantages in prediction because of the ability of processing high-dimensional feature vectors and the capability of selectively recording historical information. Compared with other advanced cost prediction methods, such as Support Vector Machine (SVM), this framework has advantages such as being able to capture long-distance dependent information and can provide short-term predictions of engineering cost indexes both effectively and accurately. This research extended current algorithm tools that can be used to forecast cost indexes and evaluated the optimization mechanism of the algorithm in order to improve the efficiency and accuracy of prediction, which have not been explored in current research knowledge.

Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the $$\mathbf {x}$$ x -dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In this paper we assume that such a transformation exists globally. However, this approach may destroy convexity in the lower level, which is very important for numerical methods. We present in this paper a solution approach for GSIP problems, which cleverly combines the above mentioned two techniques. It is shown that the convergence results for discretization methods in the case of SIP problems can be transferred to our transformation-based discretization method under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on several examples, including a problem of volume-maximal inscription of multiple variable bodies into a larger fixed body, which has never before been considered as a GSIP test problem.

The present work determines the arithmetic complexity of the index sets of u.c.e. families which are learnable according to various criteria of algorithmic learning. Specifically, we prove that the index set of codes for families that are TxtFex $$^a_b$$ -learnable is $$\Sigma _4^0$$ -complete and that the index set of TxtFex $$^*_*$$ -learnable and the index set of TxtFext $$^*_*$$ -learnable families are both $$\Sigma _5^0$$ -complete.

AHP refers to the decision making method that decomposes the elements that are always related to decision-making into levels of objectives, criteria, schemes, etc., and conducts qualitative and quantitative analysis on this basis. Fuzzy comprehensive evaluation method is a comprehensive evaluation method based on fuzzy mathematics, which converts qualitative evaluation into quantitative evaluation according to the membership theory of fuzzy mathematics. In this paper, Yaahp software is used for weight calculation and consistency test of ahp, and sensitivity analysis is also carried out. The fuzzy comprehensive evaluation method is used to determine the evaluation index set and the fuzzy comment set, and the evaluation model is constructed by fuzzy synthesis. On this basis, the empirical analysis is carried out.

The notion of T-design in a symmetric association scheme was introduced by Delsarte. A harmonic index t-design is the particular case in which the index set T consists of a single index $$\{t\}$$. Zhu et al. studied a harmonic index t-design in the binary Hamming scheme and gave a Fisher type lower bound on the cardinality. Also they defined the notion of a tight harmonic index design using the bound, and considered the classification problem. In this paper, we extend their results to the general Hamming scheme and improve their Fisher type lower bound slightly. Also using the improved Fisher type lower bound, we redefine the notion of a tight harmonic index design and consider the classification problem. Furthermore, we give a natural characterization for a harmonic index t-design and an analogue in the Hamming scheme for the construction of spherical harmonic index t-designs which was given by Bannai–Okuda–Tagami.

A compact graph-like space is a triple $(X,V,E)$ where $X$ is a compact, metrizable space, $V \subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X \setminus V \cong E \times (0,1)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.

This article considers a kind of multiple attribute group decision making problem which the decision-makers have different evaluation index set over the attribute set for all candidate alternatives. We firstly transform the multiple attribute group decision making into a fuzzy multigranulation decision making problem under the framework of different universes. We then construct a diversified binary fuzzy relation over the evaluation attribute index set associated to different decision-makers. Furthermore, using the idea of multigranulation methodology over two different universes, we define the lower and upper approximations of any fuzzy decision making object based on the diversified binary fuzzy relation. Subsequently, we establish the diversified binary fuzzy relation-based optimistic and pessimistic fuzzy multigranulation rough set over two universes. At the same time, Several interesting and important properties and conclusions are presented for the defined two types multigranulation fuzzy rough set over two universes in detail. Meanwhile, we propose an approach to multiple attribute group decision making with different evaluation index set by using the diversified binary fuzzy relation-based fuzzy multigranulation rough set over two universes. At last, a detailed comparison to the traditional studies is given and a numerical example is tested to illustrate the idea and the process of the decision making given in the new model and method.

Inspired by the paving construction method, we construct a new operation on the unit interval from a given operation defined on the unit interval and a discrete operation defined on an index set. In particular, we construct a new t-norm from a t-norm on the unit interval and a discrete t-norm; a t-conorm from a t-norm and a discrete t-superconorm; a proper uninorm from a t-norm and a discrete t-conorm and a uninorm from a t-norm and a discrete uninorm. Dually, we also construct some uninorms (including t-norms and t-conorms) from a t-conorm and a discrete t-conorm, t-subnorm, t-norm or uninorm.

We consider the semi-infinite system of polynomial inequalities of the form $$\begin{aligned} {{\mathbf {K}}}:=\{x\in {{\mathbb {R}}}^m\mid p(x,y)\ge 0,\quad \forall y\in S\subseteq {{\mathbb {R}}}^n\}, \end{aligned}$$where p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in $${{\mathbb {R}}}^n$$, $$-p(x,y)$$ is convex in x for every $$y\in S$$. We propose a procedure to construct approximate semidefinite representations of $${{\mathbf {K}}}$$. There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation sets expand and contract, respectively, and can approximate $${{\mathbf {K}}}$$ as closely as possible under some assumptions. In some special cases, we can fix one of the two indices or both. Then, we consider the optimization problem of minimizing a convex polynomial over $${{\mathbf {K}}}$$. We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of $${{\mathbf {K}}}$$. Under certain assumptions, some approximate minimizers of the optimization problem can also be obtained from the SDP relaxations. In some special cases, we show that the SDP relaxation for the optimization problem is exact and all minimizers can be extracted.

This article introduces exact testing procedures on the mean of a Gaussian process $X$ derived from the outcomes of $\ell_1$-minimization over the space of complex valued measures. The process $X$ can thought as the sum of two terms: first, the convolution between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive (centered) Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of $X$ and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation $X$ and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of $X$ in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of $X$ is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super-Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids.

In order to more accurately predict the carbon emissions of urban transportation in my country, this paper proposes a method for predicting the reduction rate of urban transportation carbon emissions based on gray correlation analysis. Based on the principle of grey relational analysis, the influencing factors of China's carbon emissions were screened. On this basis, this paper constructed the prediction model of urban traffic carbon emission reduction rate, established the evaluation index data column, determined the reference index set, and standardized the data column. Experiments show that the root mean square error of the method in this paper is less than that of the comparison method, and the average absolute percentage error is better than that of the LSSVR and ELM methods.

This paper considers the problem of multi-server Private Linear Computation, under the joint and individual privacy guarantees. In this problem, identical copies of a dataset comprised of K messages are stored on N non-colluding servers, and a user wishes to obtain one linear combination of a D-subset of messages belonging to the dataset. The goal is to design a scheme for performing the computation such that the total amount of information downloaded from the servers is minimized, while the privacy of the D messages required for the computation is protected. When joint privacy is required, the index set of these D messages must be kept private, and when individual privacy is required, the identity of each individual message required for the computation must be kept private. In this work, we characterize the capacity, which is defined as the maximum achievable download rate, under both joint and individual privacy requirements. Our converse proofs are based on reduction from two variants of the multi-server Private Information Retrieval problem with side information. Our achievability schemes build up on our recently proposed schemes for single-server Private Linear Transformation and the multi-server private computation scheme proposed by Sun and Jafar. Using similar techniques, we also establish bounds on the capacity for the cases in which the user wants to compute multiple (more than one) linear combinations of a D-subset of messages.