Class Of Cones Research Articles

Stability of bounded subsets of Metzler sparse matrix cones

Abstract We study the existence of Hurwitz matrices in a class of Metzler sparse matrix cones, where the non-zero entries are constrained to be in a bounded set. In particular, we provide necessary and sufficient conditions for the existence of one or more Hurwitz matrices in this class of sparse matrix spaces, for scenarios with uniform and non-uniform bounds. Several examples illustrate our results.

Linear Satisfiability Preserving Assignments

In this paper, we study several classes of satisfiability preserving assignments to the constraint satisfaction problem (CSP). In particular, we consider fixable, autark and satisfying assignments. Since it is in general NP-hard to find a nontrivial (i.e., nonempty) satisfiability preserving assignment, we introduce linear satisfiability preserving assignments, which are defined by polyhedral cones in an associated vector space. The vector space is obtained by the identification, introduced by Kullmann, of assignments with real vectors. We consider arbitrary polyhedral cones, where only restricted classes of cones for autark assignments are considered in the literature. We reveal that cones in certain classes are maximal as a convex subset of the set of the associated vectors, which can be regarded as extensions of Kullmann's results for autark assignments of CNFs. As algorithmic results, we present a pseudo-polynomial time algorithm that computes a linear fixable assignment for a given integer linear system, which implies the well known pseudo-polynomial solvability for integer linear systems such as two-variable-per-inequality (TVPI), Horn and q-Horn systems.

Delayed cone-opponent signals in the luminance pathway.

Cone signals in the luminance or achromatic pathway were investigated by measuring how the perceptual timing of M- or L-cone-detected flicker depended on temporal frequency and chromatic adaptation. Relative timings were measured, as a function of temporal frequency, by superimposing M- or L-cone-isolating flicker on "equichromatic" flicker (flicker of the same wavelength as the background) and asking observers to vary contrast and phase to cancel the perception of flicker. Measurements were made in four observers on up to 35 different backgrounds varying in wavelength and radiance. Observers showed substantial perceptual delays or advances of L- and M-cone flicker that varied systematically with cone class, background wavelength, and radiance. Delays were largest for M-cone-isolating flicker. Although complex, the results can be characterised by a surprisingly simple model in which the representations of L- and M-cone flicker are comprised not only of a fast copy of the flicker signal, but also of a slow copy that is delayed by roughly 30 ms and varies in strength and sign with both background wavelength and radiance. The delays, which are too large to be due to selective cone adaptation by the chromatic backgrounds, must arise postreceptorally. Clear evidence for the slow signals can also be found in physiological measurements of horizontal and magnocellular ganglion cells, thus placing the origin of the slow signals in the retina-most likely in an extended horizontal cell network. Luminance-equated stimuli chosen to isolate chromatic channels may inadvertently generate slow signals in the luminance channel.

Transfer learning for one class SVM adaptation to limited data distribution change

Data based one class classification rules are widely used in system monitoring. Due to maintenance for example, we may come across a change of data distribution with respect to training data. While lacking of representative samples for the new data set, one can try to adapt the former learned detection rule to the new data set instead of retraining a new rule which implies to gather a significant amount of data. Based on the above, a multi-task learning detection rule approach is proposed to deal with the training of the updated system as some new data are available. The key feature of the new approach is the introduction of a parameter to control how much we rely on the former model. This parameter has to be set and changed as the amount of new data coming from the system increases. We define the new detection model as a classical one class SVM with a specific kernel matrix which depends on the parameter we introduced. A kernel adaptation method for C-one class SVM is developed in order to get the path solution along that parameter and a criteria is established to select a good value. Experiments conducted on toy data and real data set show that the proposed method could adapt to data change, and it gives a good transition from the old detection rule to the new one which is just obtained using the new data set only when the number of samples gathered from that new one is large enough.

Mapping cones and separable states

We study mapping cones and their dual cones of positive maps of the $$n\times n$$ matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular the composition of a map from the cone with a map in the dual cone is super-positive, and so the natural state it defines is separable.

Cones with semi-interior points and equilibrium

We study exchange economies in ordered normed spaces (X,‖⋅‖) where agents have possibly different consumption sets. We define the notion of semi-interior point of the positive cone X+ of X, a notion weaker than the one of interior point and we study the existence of equilibrium in the case where X+ has semi-interior points. In Section 4, we study the case where X+ has interior points and we prove a second welfare theorem and the existence of equilibrium. Subsequently we apply these results in the case where X+ has semi-interior points. In the case of semi-interior points the supporting price vectors are continuous with respect to a new norm ∣∣∣⋅∣∣∣ on X which is strongly related with the initial norm and the ordering, and in some sense can be considered as an extension of the norm adopted in classical equilibrium models. Many examples of cones in normed and Banach spaces with semi-interior points but with empty interior are provided, showing that this class of cones is a rich one. In the last section we apply our results to strongly reflexive cones.

Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming

In conic linear programming -- in contrast to linear programming -- the Lagrange dual is not an exact dual: it may not attain its optimal value, or there may be a positive duality gap. The corresponding Farkas' lemma is also not exact (it does not always prove infeasibility). We describe exact duals, and certificates of infeasibility and weak infeasibility for conic LPs which are nearly as simple as the Lagrange dual, but do not rely on any constraint qualification. Some of our exact duals generalize the SDP duals of Ramana, and Klep and Schweighofer to the context of general conic LPs. Some of our infeasibility certificates generalize the row echelon form of a linear system of equations: they consist of a small, trivially infeasible subsystem obtained by elementary row operations. We prove analogous results for weakly infeasible systems. We obtain some fundamental geometric corollaries: an exact characterization of when the linear image of a closed convex cone is closed, and an exact characterization of nice cones. Our infeasibility certificates provide algorithms to generate {\em all} infeasible conic LPs over several important classes of cones; and {\em all} weakly infeasible SDPs in a natural class. Using these algorithms we generate a public domain library of infeasible and weakly infeasible SDPs. The status of our instances can be verified by inspection in exact arithmetic, but they turn out to be challenging for commercial and research codes.

Rapid Recovery of Visual Function Associated with Blue Cone Ablation in Zebrafish.

Hurdles in the treatment of retinal degeneration include managing the functional rewiring of surviving photoreceptors and integration of any newly added cells into the remaining second-order retinal neurons. Zebrafish are the premier genetic model for such questions, and we present two new transgenic lines allowing us to contrast vision loss and recovery following conditional ablation of specific cone types: UV or blue cones. The ablation of each cone type proved to be thorough (killing 80% of cells in each intended cone class), specific, and cell-autonomous. We assessed the loss and recovery of vision in larvae via the optomotor behavioural response (OMR). This visually mediated behaviour decreased to about 5% or 20% of control levels following ablation of UV or blue cones, respectively (P<0.05). We further assessed ocular photoreception by measuring the effects of UV light on body pigmentation, and observed that photoreceptor deficits and recovery occurred (p<0.01) with a timeline coincident to the OMR results. This corroborated and extended previous conclusions that UV cones are required photoreceptors for modulating body pigmentation, addressing assumptions that were unavoidable in previous experiments. Functional vision recovery following UV cone ablation was robust, as measured by both assays, returning to control levels within four days. In contrast, robust functional recovery following blue cone ablation was unexpectedly rapid, returning to normal levels within 24 hours after ablation. Ablation of cones led to increased proliferation in the retina, though the rapid recovery of vision following blue cone ablation was demonstrated to not be mediated by blue cone regeneration. Thus rapid visual recovery occurs following ablation of some, but not all, cone subtypes, suggesting an opportunity to contrast and dissect the sources and mechanisms of outer retinal recovery during cone photoreceptor death and regeneration.

Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones

The Koszul–Vinberg characteristic function plays a fundamental role in the theory of convex cones. We give an explicit description of the function and related integral formulas for a new class of convex cones, including homogeneous cones and cones associated with chordal (decomposable) graphs appearing in statistics. Furthermore, we discuss an application to maximum likelihood estimation for a certain exponential family over a cone of this class.

Warped products admitting a curvature bound

This paper completes a fundamental construction in Alexandrov geometry. Previously we gave a new construction of metric spaces with curvature bounds either above or below, namely warped products with intrinsic metric space base and fiber, and with possibly vanishing warping functions – thereby extending the classical cone and suspension constructions from interval base to arbitrary base, and furthermore encompassing gluing constructions. This paper proves the converse, namely, all conditions of the theorems are necessary. Note that in the cone construction, both the construction and its converse are widely used. We also show that our theorems for curvature bounded above and below, respectively, are dual. We give the first systematic development of basic properties of warped products of metric spaces with possibly vanishing warping functions, including new properties.

Vector optimization over cones involving support functions using generalized $$(\Phi ,\rho )$$ ( Φ , ρ ) -convexity

In this paper we introduce new classes of cone-\((\Phi ,\rho )\)-convex functions which extend several generalized convexity notions introduced in the literature. We then employ the defined functions to study optimality and duality results for a vector optimization problem over cones involving support functions. A numerical example is given to illustrate the notions introduced and the results proved.

Spectral sensitivity of guppy visual pigments reconstituted in vitro to resolve association of opsins with cone cell types

The guppy (Poecilia reticulata) shows remarkable variation of photoreceptor cells in the retina, especially those sensitive to middle-to-long wavelengths of light. Microspectrophotometry (MSP) has revealed varying “green”, “green-yellow” and “yellow” cone cells among guppies in Trinidad and Venezuela (Cumana). In the guppy genome, there are four “long-wave” opsin loci (LWS-1, -2, -3 and -4). Two LWS-1 alleles have potentially differing spectral sensitivity (LWS-1/180Ser and LWS-1/180Ala). In addition, two “middle-wave” loci (RH2-1 and -2), two “short-wave” loci (SWS2-A and -B), and a single “ultraviolet” locus (SWS1) as well as a single “rhodopsin” locus (RH1) are present. However, the absorption spectra of these photopigments have not been measured directly and the association of cell types with these opsins remains speculative. In the present study, we reconstituted these opsin photopigments in vitro. The wavelengths of maximal absorbance (λmax) were 571nm (LWS-1/180Ser), 562nm (LWS-1/180Ala), 519nm (LWS-3), 516nm (LWS-2), 516nm (RH2-1), 476nm (RH2-2), 438nm (SWS2-A), 408nm (SWS2-B), 353nm (SWS1) and 503nm (RH1). The λmax of LWS-3 is much shorter than the value expected (560nm) from the “five-sites” rule. The two LWS-1 alleles could explain difference of the reported MSP λmax values for the yellow cone class between Trinidad and Cumana guppies. Absence of the short-wave-shifted LWS-3 and the green-yellow cone in the green swordtail supports the hypothesis that this cell class of the guppy co-expresses the LWS-1 and LWS-3. These results reveal the basis of variability in the guppy visual system and provide insight into the behavior and ecology of these tropical fishes.

Resource Efficiency: A New Beamforming Design for Multicell Multiuser Systems

Apart from spectral efficiency (SE), energy efficiency (EE) has become an important metric for future wireless communication systems. However, how to achieve balance between SE and EE for a multicell multiuser coordinated beamforming system remains an open problem. In this paper, we propose a new paradigm for EE–SE balancing, namely, using a new metric called resource efficiency (RE) as the objective function to achieve a reasonable tradeoff between SE and EE. The considered optimization problem is more complex than traditional problems and, in general, is NP-hard for finding the globally optimal solution. To obtain insight into the problem, we first equivalently transform the original problem into a parameterized subtractive form optimization problem with some auxiliary variables by using the fraction programming theory. Then, a successive convex approximation method and the classical second-order cone programming method are jointly used to solve the equivalent problem. Finally, an effective algorithm with proven convergence is proposed to obtain the solution to the considered RE optimization. In addition, the algorithm is further extended to consider imperfect channel state information (CSI) using the worst-case design. Extensive numerical simulations are given to validate the effectiveness of our algorithm.

HOMOGENEOUS SELF-DUAL METHODS FOR SYMMETRIC CONES UNDER UNCERTAINTY

Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space have been proposed by Jin et al. in [17]. Alzalg [8] has adopted their work to derive homogeneous selfdual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these to derive homogeneous selfdual algorithms for stochastic programs with finite event space over the much wider class of all symmetric cones. They include among others, stochastic semidefinite programs and stochastic second-order cone programs.

A Spectral Estimation Theory for Color Appearance Matching

There are several reports describing color vision in subjects who are dichromatic in one eye and trichromatic in the other. Formulae fit to the between-eye color appearance matches in such unilateral dichromats have been used to predict color appearance for dichromats [1][2]. In this paper, we describe a general principle, spectral estimation theory, that guides how to predict the mapping from two cone class absorptions in a dichromatic eye to three cone class absorptions in a trichromatic eye. The theory predicts matches by first estimating the smoothest, non-negative, spectral power distribution that is consistent with the measured cone absorption rates (e.g., of a dichromat). We then use this spectral estimate to calculate the absorption rate for the missing cone type (the standard color observer). In addition to predicting color appearance of dichromats, the theory predicts color appearance matches between color anomalous subjects and the standard color observer. Finally, the theory offers guidance about the possible effects on color appearance of gene therapy treatment for colorblindness.

Isac’s Cones

This is a very short research work representing an homage to the regretted Professor George Isac, Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. 17000, Kingston, Ontario, Canada, K7K 7B4. Professor Isac introduced the notion of “nuclear cone” in 1981, published in 1983 and called later as “supernormal cone” since it appears stronger than the usual concept of “normal cone”. For the first time, we named these convex cones as “Isac’s Cones” in 2009 , after the acceptance on professor Isac’s part. This study is devoted to Isac’s cones, including significant examples, comments and several pertinent references, with the remark that this notion has its real place in Hausdorff locally convex spaces not in the normed linear spaces, having strong implications and applications in the efficiency and optimization. Isac’s cones represent the largest class of convex cones discovered till now in separated locally convex spaces ensuring the existence and important properties for the efficient points under completeness instead of compactness.

Is adding a new class of cones to the retina sufficient to cure color-blindness?

New genetic methods have made it possible to substitute cone pigments in the retinas of adult nonhuman primates. Doing so influences the animals' visual abilities, demonstrating that the gene therapy was effective. However, we argue that no studies conducted so far have unambiguously demonstrated that the experimental animals have also acquired the ability to make new color distinctions. Simply put, it has been shown that animals that underwent the gene treatment can now-in addition to finding a red ball on a grayish background-find a green ball on a grayish background. However, it has not been shown that the animals can distinguish a red ball from a green one. For most people, that essential ability would be the primary reason for wanting to undergo a treatment for color-blindness in the first place, for instance, because their color-blindness currently prevents them from pursuing a career as a pilot or firefighter. It is important to point out such possible limitations of gene therapy for color-blindness to avoid unwarranted expectations in both clinicians and patients. To explain the origin of our concerns, we simulate how replacing the pigment of some cones is expected to influence the outcomes on the behavioral test used so far. The simulations show that this test does not provide conclusive evidence that the animals acquired the ability to make new chromatic distinctions. In our view, it is therefore premature to claim that human color-blindness can be cured through gene therapy. We propose a test that would provide more conclusive evidence of fundamentally altered color vision after gene therapy.

Targeted Ablation of the Pde6h Gene in Mice Reveals Cross-species Differences in Cone and Rod Phototransduction Protein Isoform Inventory

Phosphodiesterase-6 (PDE6) is a multisubunit enzyme that plays a key role in the visual transduction cascade in rod and cone photoreceptors. Each type of photoreceptor utilizes discrete catalytic and inhibitory PDE6 subunits to fulfill its physiological tasks, i.e. the degradation of cyclic guanosine-3',5'-monophosphate at specifically tuned rates and kinetics. Recently, the human PDE6H gene was identified as a novel locus for autosomal recessive (incomplete) color blindness. However, the three different classes of cones were not affected to the same extent. Short wave cone function was more preserved than middle and long wave cone function indicating that some basic regulation of the PDE6 multisubunit enzyme was maintained albeit by a unknown mechanism. To study normal and disease-related functions of cone Pde6h in vivo, we generated Pde6h knock-out (Pde6h(-/-)) mice. Expression of PDE6H in murine eyes was restricted to both outer segments and synaptic terminals of short and long/middle cone photoreceptors, whereas Pde6h(-/-) retinae remained PDE6H-negative. Combined in vivo assessment of retinal morphology with histomorphological analyses revealed a normal overall integrity of the retinal organization and an unaltered distribution of the different cone photoreceptor subtypes upon Pde6h ablation. In contrast to human patients, our electroretinographic examinations of Pde6h(-/-) mice suggest no defects in cone/rod-driven retinal signaling and therefore preserved visual functions. To this end, we were able to demonstrate the presence of rod PDE6G in cones indicating functional substitution of PDE6. The disparities between human and murine phenotypes caused by mutant Pde6h/PDE6H suggest species-to-species differences in the vulnerability of biochemical and neurosensory pathways of the visual signal transduction system.

The Isbell monad

In 1966 [7], John Isbell introduced a construction on categories which he termed the “couple category” but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2-category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system [5] from orthogonal classes of arrows to orthogonal classes of cocones and cones.

$p$-variations of vector measures with respect to vector measures and integral representation of operators

In this paper we provide two representation theorems for two relevant classes of operators from any $p$-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite $p$-semivariation and $p$-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures.