Quotient Problem Research Articles

The Relationship Between Emotional Intelligence and Adversity Quotient in Working Students

This research is motivated by the phenomenon of the adversity quotient problem in working students which is characterized by a lack of enthusiasm for participating in the lecture process according to the adversity quotient aspect, namely endurance. In this case, the resilience of students to work is not reflected in facing their difficulties. Then often complain in lecture activities, including the aspect of the adversity quotient, namely control. This makes self-control in working students low, where working students cannot carry out their duties as working students. The purpose of this study was to determine the relationship between emotional intelligence and the adversity quotient of working students at the Muhammadiyah University of Sidoarjo. This research is a correlational quantitative research. The sample in this study was 251 students who were taken using the Incidental Sampling technique. The data collection technique used a psychological scale of emotional intelligence and adversity quotient which was arranged on a Likert scale. The data analysis technique for hypothesis testing was carried out with the Pearson Correlaton Product Moment. The results of this study indicate the correlation coefficient (rxy) = 0.411 (significance 0.000 <0.05). This means that there is a positive relationship between emotional intelligence and adversity quotient in student workers. The higher the emotional intelligence, the higher the adversity quotient, and vice versa. The determination test shows the effect of emotional intelligence on the adversity quotient of 16.8%.

Revisiting Mazur separable quotient problem (1932)

The centenary of the still open problem of the existence of a separable quotient of infinite dimension in a Banach space $ X $ will be in 2032. This note aims to give a quick overview of this problem, which it is known that it has a positive solution for Banach spaces with, rough speaking, 'large density character' and also with 'small density character'. Proofs of several results have been reproduced or simplified to motivate interest in solving this so-called 'Mazur separable quotient Problem'.

Extrapolated Proximal Subgradient Algorithms for Nonconvex and Nonsmooth Fractional Programs

In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, which encompass many important modern optimization problems arising from diverse areas such as the recently proposed scale-invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and that any one of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted fast iterative shrinking-threshold algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka–Łojasiewicz property. Our algorithm exhibits linear convergence for the scale-invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. When the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose another extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples.

Tensor local linear embedding with global subspace projection optimisation

In this paper, a novel tensor dimensionality reduction (TDR) approach is proposed, which maintains the local geometric structure of tensor data by tensor local linear embedding and explores the global feature by optimising global subspace projection. Firstly, we analyse the local linear feature of tensor data for learning the linear separable embedding of the tenor data. Furthermore, a global subspace projection distance minimisation strategy is introduced to extract the global characteristic of the tensor data. The aim of this strategy is to find an optimal low-dimensional subspace for TDR. In particular, two novel TDR algorithms are developed by the ensemble of tensor local feature preservation and global subspace projection distance minimisation, which express the subspace projection optimisation as an iteration optimisation problem and a Rayleigh quotient problem, respectively. The extensive experimental results on tensor data classification and clustering have demonstrated the proposed algorithms performed well.

Nonlinear estimation based on conversion-sample optimization

For nonlinear estimation, the linear minimum mean square error (LMMSE) estimator using the original measurement is the best of all estimators linear in the measurement. Recently, it has been proven that an LMMSE estimator using a stacked measurement vector composed of the original measurement and its nonlinear conversion (transformation or function) outperforms the original LMMSE estimator. However, to find an optimal conversion for various nonlinear problems is very difficult, because it involves functional optimization. Recognizing that the conversion affects estimation performance only through conversion-dependent quantities needed by the augmented LMMSE-based estimator, this paper proposes an optimized conversion-sample filter (OCF), which optimizes all these quantities to optimize estimation performance. These quantities are much easier to obtain than the optimal conversion itself. Using deterministic sampling, obtaining these quantities amounts to determining the converted values at both the deterministic sample points and the measurement value. In the OCF, these values are obtained optimally and analytically by solving a constrained Rayleigh quotient problem. The proposed OCF has a simple and analytical form and can be generally applied to nonlinear estimation. Simulation results demonstrate the effectiveness of the proposed OCF compared with some popular nonlinear estimators, including several LMMSE-based filters, a recently-proposed uncorrelated conversion based filter, and a particle filter.

Hybrid Precoding for Massive MIMO With Low Rank Channels: A Two-Stage User Scheduling Approach

To fully reap the benefits of massive multiple-input multiple-output hybrid analog and digital precoding in frequency division duplexing single-cell systems, a two-stage precoder is developed utilizing the signal-to-leakage-plus-noise ratio metric. The main idea of this technique is to jointly design the analog precoder based only on the long-term channel statistics information at the transmitter, i.e., the channel mean and reconstructed reduced rank covariance statistics, while the digital precoder is designed based on the instantaneous channel state information of the reduced dimensionality effective reconstructed channel. Consequently, we can significantly reduce the downlink training and uplink feedback overhead analogously to the rank of the resultant effective channel. The two extremes of full channel state information at the transmitter (CSIT) and statistical CSIT are also investigated. The performance gap between the full and statistical CSIT corroborates the importance of the proposed two-stage CSIT approach. These precoders are then extended to multi-cell systems. It is shown that the digital baseband precoder design problem reduces to the generalized Rayleigh quotient problem, while the analog precoder design problem reduces to the quotient trace problem, also known as the ratio trace problem. These dimensionality reduction problems are solved via the generalized eigenvalue decomposition method. Finally, in the presence of multiuser diversity where only a subset of the users are scheduled, to considerably alleviate the channel estimation and feedback overhead burden, a low-complexity one-stage and two-stage CSIT joint user scheduler and precoder algorithms are developed.

The Tubby Torus as a Quotient Group

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

Observations on the Separable Quotient Problem for Banach Spaces

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.

Open book decompositions of links of quotient surface singularities and support genus problem

In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singularities. We show that for many types of the quotient surface singularities the Milnor genus is equal to the support genus. In the remaining cases we are able to find a small upper bound for the support genus.

The separable quotient problem for topological groups

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces.For a topological group G there are four natural analogous problems: Does G have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. Positive answers to all four questions are proved for groups G which belong to the important classes of (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups.However, a surprising example of an uncountable precompact group G is produced which has no non-trivial separable quotient group other than the trivial group. Indeed Gτ has the same property, for every cardinal number τ ≥ 1.

A REMARK ON THE SEPARABLE QUOTIENT PROBLEM FOR TOPOLOGICAL GROUPS

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

Multiuser Interference Cancellation for GFDM With Timing and Frequency Offsets

In uplink multiuser systems, different timing offset (TO) and/or carrier frequency offset (CFO) of multiple users cause multiuser interference (MUI) to a desired user. In this paper, we propose two different MUI cancellation methods, namely weighted parallel interference cancellation (WPIC) and adaptive interference cancellation filter (AICF), for uplink transmission with generalized frequency division multiplexing (GFDM) waveform over Rayleigh fading channel given the existence of TO and CFO of multiple users. The proposed WPIC performs MUI estimation using the detected data symbol of other users and obtains the optimal weight to maximize the signal-to-interference ratio (SIR) of the desired user. We also propose the AICF that aims to maximize the SIR of the desired user. Based on our SIR expression derived in closed-form in the presence of TO and CFO, the AICF is obtained by solving the Rayleigh quotient problem. Lastly, the impact of multiple antennas at the receiver on MUI cancellation is also analyzed to improve reliability. The simulation results show that the proposed WPIC and AICF effectively eliminate MUI and that the AICF can outperform the WPIC in terms of the SIR and spectral efficiency. In addition, the multiple receiver antennas effectively cancel MUI as the number of antennas goes to infinity.

A topological group observation on the Banach–Mazur separable quotient problem

The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, Tω, where T denotes the compact unit circle group. Indeed it is shown that every locally convex space, which has a subspace which is an infinite-dimensional Fréchet space, has Tω as a quotient group.

Separability of Topological Groups: A Survey with Open Problems

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

On the separable quotient problem for Banach spaces

While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of infinite-dimensional separable quotients in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the presented results are consequences of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces $c_{0}$ and $\ell _{p}$, for $1\leq p\leq \infty $. Also recent results of Argyros, Dodos, Kanellopoulos [1] and Śliwa [64] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica.

The Quotient Problem for Entire Functions

AbstractLet $\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and $\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set ${\mathcal{N}}$ of natural numbers such that their ratio $\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and $G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the $f_{i}$ and $g_{j}$ are nonconstant entire functions and the $a_{i}$ and $b_{j}$ are non-zero constants except that $a_{0}$ can be zero. We will show that the set ${\mathcal{N}}$ of natural numbers such that $F(n)/G(n)$ is an entire function is finite under the assumption that $f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set $(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

The uniform bounded deciding property and the separable quotient problem

Saxon–Wilansky’s paper The equivalence of some Banach space problems contains six properties equivalent to the existence of an infinite dimensional separable quotient in a Banach space with nice simplified proofs. In the frame of uniform bounded deciding property, we prove that for an infinite dimensional Banach space $$(E,\left\| \cdot \right\| )$$ the following properties are equivalents: 1) The unit sphere $$S_{E}$$ contains a dense and non uniform bounded deciding subset. 2) The unit sphere $$S_{E}$$ contains a dense and non strong norming subset. 3) $$(E,\left\| \cdot \right\| )$$ admits an infinite dimensional separable quotient.

Separable quotients for less-than-barrelled function spaces

We solve the separable quotient problem for spaces Cp(X) and Cc(X) that are dual locally complete (dlc), greatly extending the positive solution for barrelled Cc(X). Indeed, ours is either the best or next-to-best weak barrelledness extension possible. Yet some interesting non-dlc spaces Cp(X) and Cc(X) also admit separable quotients.

Secure beamforming for two-way multiantenna relay systems

In two-way multiantenna relay systems, confidential information can be easily intercepted by eavesdroppers. To overcome this problem, two secure beamforming schemes are proposed: secrecy sum rate maximization beamforming (SSRMB) and null space beamforming (NSB). For the SSRMB scheme, the nonconvex secrecy sum rate maximization problem is solved using a successive convex approximation strategy, which is convergent. The obtained beamforming solution is locally optimal to the original problem. For the special scenario where the number of antennas at the relay is more than the number of eavesdroppers, the NSB scheme with a low complexity is proposed. The beamforming solution is obtained by solving a generalized Rayleigh quotient problem. Simulation results demonstrate that the proposed schemes can effectively improve the security performance, and the SSRMB scheme has a fast convergence rate.

Efimov spaces and the separable quotient problem for spaces Cp(K)

The classic Rosenthal–Lacey theorem asserts that the Banach space C(K) of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ℓ2. More recently, Ka̧kol and Saxon [20] proved that the space Cp(K) endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence Cp(βN) lacks infinite-dimensional separable quotient algebras. This motivates the following question: (⁎) DoesCp(K)admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does Cp(βN) admit SQ? Our main theorem implies that Cp(K) has SQ for any compact space K containing a copy of βN. Consequently, this result reduces problem (⁎) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ◊), that for some Efimov space K the space Cp(K) has SQ. Some applications of the main result are provided.