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Let $$k\ge 2$$ be an integer and $$S=\{1\}\cup S'$$ be a Goldbach-type set such that $$S'\subseteq 2{\mathbb {N}}$$ and $$\{2,4,6,8,12,18\}\in S'$$ . If a multiplicative function f satisfies $$\begin{aligned} f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k) \end{aligned}$$ for arbitrary $$x_1,\ldots ,x_k\in S$$ , then f is the identity function $$f(n)=n$$ for all $$n\in {\mathbb {N}}$$ . In particular, the set of practical numbers is a k-additive uniqueness set.

We prove that the set $$\mathcal {P}$$ ( $$\mathcal {H}$$ , resp.) of all positive pentagonal (hexagonal, resp.) numbers is an additive uniqueness set for the collection of multiplicative functions; if a multiplicative function f satisfies the equation $$\begin{aligned} f(a+b) = f(a) + f(b) \end{aligned}$$ for all $$a, b \in \mathcal {P}$$ ( $$\mathcal {H}$$ , resp.), then f is the identity function.

Overfitting frequently occurs in deep learning. In this paper, we propose a novel regularization method called drop-activation to reduce overfitting and improve generalization. The key idea is to drop nonlinear activation functions by setting them to be identity functions randomly during training time. During testing, we use a deterministic network with a new activation function to encode the average effect of dropping activations randomly. Our theoretical analyses support the regularization effect of drop-activation as implicit parameter reduction and verify its capability to be used together with batch normalization (Ioffe and Szegedy in Batch normalization: accelerating deep network training by reducing internal covariate shift. arXiv:1502.03167 , 2015). The experimental results on CIFAR10, CIFAR100, SVHN, EMNIST, and ImageNet show that drop-activation generally improves the performance of popular neural network architectures for the image classification task. Furthermore, as a regularizer drop-activation can be used in harmony with standard training and regularization techniques such as batch normalization and AutoAugment (Cubuk et al. in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 113–123, 2019). The code is available at https://github.com/LeungSamWai/Drop-Activation .

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$ , the identity function ${\mathbf {x}}$ , and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^{\mathbf {x}}}$ . Here we prove that the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $ . As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$ . We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$ , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations.

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function., v3: correct typo (after publication), v2: minor changes, to appear in Math. Scand., v1: 5 pages

Employing the concept and function of tangency with similarity measures and counterpart distances for reliable medical consultations has been extensively studied in the past decades and results in lots of isomorphic measures for application. We compared the majority of such isomorphic measures proposed by various researchers and classified them into (a) maximum norm and (b) one-norm categories. Moreover, we found that previous researchers used monotonic functions to transform an identity function and resulted in complicated expressions. In this study, we provide a theoretical foundation to explain the isomorphic nature of a newer measure proposed by the following research paper against its studied existing one in deriving the same pattern recognition results. Specifically, this study initially proposes two similarity measures using maximum norm, arithmetic mean, and aggregation operators and followed by a detailed discussion on their mathematical characteristics. Subsequently, a simplified version of such measures is presented for easy application. This study completely covers two previous methods to point out that the complex approaches used were unnecessary. The findings will help physicians, patients, and their family members to obtain a proper medical diagnosis during multiple examinations.

Let be a metric space and be a continuous map. The notion of the -average shadowing property ( ASP ) for a continuous map on –space is introduced and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if has ASP, then has ASP for every . We prove that if a map be pseudo-equivariant with dense set of periodic points and has the ASP, then is weakly mixing. We also show that if is a expansive pseudo-equivariant homeomorphism that has the ASP and is topologically mixing, then has a -specification. We obtained that the identity map on has the ASP if and only if the orbit space of is totally disconnected. Finally, we show that if is a pseudo-equivariant map, and the trajectory map is a covering map, then has the ASP if and only if the induced map has ASP.

We present a short-step interior-point algorithm (IPA) for sufficient linear complementarity problems (LCPs) based on a new search direction. An algebraic equivalent transformation (AET) is used on the centrality equation of the central path system and Newton’s method is applied on this modified system. This technique was offered by Zsolt Darvay for linear optimization in 2002. Darvay first used the square root function as AET and in 2012 Darvay et al. applied this technique with an other transformation formed by the difference of the identity map and the square root function. We apply the AET technique with the new function to transform the central path equation of the sufficient LCPs. This technique leads to new search directions for the problem. We introduce an IPA with full Newton steps and prove that the iteration bound of the algorithm coincides with the best known one for sufficient LCPs. We present some numerical results to illustrate performance of the proposed IPA on two significantly different sets of test problems and compare it, with related, quite similar variants of IPAs.

We construct the first example of a $C^*$-algebra $A$ with the properties in the title. This gives a new example of non-nuclear $A$ for which there is a unique $C^*$-norm on $A \otimes A^{op}$. This example is of particular interest in connection with the Connes-Kirchberg problem, which is equivalent to the question whether $C^*({\bb F}_2)$, which is known to have the LLP, also has the WEP. Our $C^*$-algebra $A$ has the same collection of finite dimensional operator subspaces as $C^*({\bb F}_2)$ or $C^*({\bb F}_\infty)$. In addition our example can be made to be quasidiagonal and of similarity degree (or length) 3. In the second part of the paper we reformulate our construction in the more general framework of a $C^*$-algebra that can be described as the \emph{limit both inductive and projective} for a sequence of $C^*$-algebras $(C_n)$ when each $C_n$ is a \emph{subquotient} of $C_{n+1}$. We use this to show that for certain local properties of injective (non-surjective) $*$-homomorphisms, there are $C^*$-algebras for which the identity map has the same properties as the $*$-homomorphisms., We recommend the shorter initial version for a first reading. Final version v3 will appear in Inventiones Math. v4 contains a more detailed end of proof of Lemma 7.1. v5 clarifies certain ambiguities (mainly in Lemma 5.3 and Th. 9.2) unfortunately overlooked in the (now published) version v3

The ability to sustain indirect reciprocity is an example of collective intelligence. It is increasingly relevant to future technology and autonomous machines that need to function in a coalition. Indirect reciprocity involves providing benefit to others without guaranteeing a future return. The identity through which an agent presents itself to others is fundamental, as this is how the reputation of an agent is considered. In this paper, we examine the sharing of identity between agents, which is an important and frequently overlooked issue when considering indirect reciprocity. We model an agent's identity using traits, which can be shared with other agents, and offer a basis for an agent to change their identity. Through this approach, we determine how shared identity affects cooperation, and the conditions through which cooperation can be sustained. This also helps us to understand how and why behavioural strategies involving identity function are put in place, such as whitewashing. The framework offers the opportunity to assess the interplay between the sharing of traits and the cost, in terms of reduced cooperation and opportunities for shirkers to benefit.

In nature, homologues can be seen in structures of common evolutionary origin, which are composed of similar elements in the same zones. Homology shows the relationship of living organisms, which may differ externally, but their structure (bone and muscle) is characterized by significant similarity. The equivalent of homologous transformations in mathematics is homeomorphism, which is one of the basic concepts of the department of contemporary mathematics dealing with the study of the properties of topological geometric figures and solids (properties which do not change even after radical continuous deformation of figures), called topology. Homeomorphism is about transforming figures without tearing them apart and gluing them together, only by stretching, compressing, bending and twisting them, which enables continuity and identity functions to be maintained. Homeomorphic spaces are indistinguishable from the point of view of topology. Homeomorphism of space ensures the preservation of topological invariables, which include a number of features such as closeness, openness, compactness, centrality or cohesion. Optimization searches for solids based on topological transformations, carried out in the scope of architectural and structural criteria can be an important element of analyses in the interdisciplinary design process. On their basis it is possible to make a conscious choice between individual forms inspired by geometry or structures found in nature. The paper presents the results of model analyses of structural forms formed on the basis of topological transformations with determination of their effectiveness in terms of minimum mass. Optimization searches due to the minimum mass are an important element from the design of objects due to the idea of sustainable development.

Graham Priest has recently proposed a solution to the problem of the One and the Many which involves inconsistent objects and a non-transitive identity relation. We show that his solution entails either that the object everything is identical with the object nothing or that they are mutual parts; depending on whether Priest goes for an extensional or a non-extensional mereology.

We prove that a $$C^k$$, $$k\ge 2$$ pseudo-rotation f of the disc with non-Brjuno rotation number is $$C^{k-1}$$-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its $$C^1$$ norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Russmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

An algebraic semantics, based on factor algebras, for one-way and two-way implicative verbs is proposed. Implicative verbs denote elements of filters or of ideals generated by identity functions in factor algebras. This semantics explains in particular the problem of implicational equivalence raised by two-way implicative verbs, and shows that the negation necessary to establish the implicativity of these verbs is the negation which preserves the presuppositions of sentences with implicative verbs. In addition, it follows from the proposed semantics that any two implicative verbs denoting in the same algebra but belonging to different categories, are semantically related.

This article proposes a universal activation function (UAF) that achieves near optimal performance in quantification, classification, and reinforcement learning (RL) problems. For any given problem, the gradient descent algorithms are able to evolve the UAF to a suitable activation function by tuning the UAF’s parameters. For the CIFAR-10 classification using the VGG-8 neural network, the UAF converges to the Mish like activation function, which has near optimal performance $$F_{1}=0.902\pm 0.004$$ F 1 = 0.902 ± 0.004 when compared to other activation functions. In the graph convolutional neural network on the CORA dataset, the UAF evolves to the identity function and obtains $$F_1=0.835\pm 0.008$$ F 1 = 0.835 ± 0.008 . For the quantification of simulated 9-gas mixtures in 30 dB signal-to-noise ratio (SNR) environments, the UAF converges to the identity function, which has near optimal root mean square error of $$0.489\pm 0.003~\mu {\mathrm{M}}$$ 0.489 ± 0.003 μ M . In the ZINC molecular solubility quantification using graph neural networks, the UAF morphs to a LeakyReLU/Sigmoid hybrid and achieves RMSE=$$0.47\pm 0.04$$ 0.47 ± 0.04 . For the BipedalWalker-v2 RL dataset, the UAF achieves the 250 reward in $${961\pm 193}$$ 961 ± 193 epochs with a brand new activation function, which gives the fastest convergence rate among the activation functions.

We develop techniques for determining certain structural properties of inverse limits on compacta. In particular, we show if easily observable properties of the bonding mappings are present, then one can identify nested sequences of subcompacta of the inverse limit space whose members are copies of subcompacta of the factor spaces, and a sequence of retractions of the inverse limit space onto members of this nested sequence that converges uniformly to the identity mapping. We discuss applications to inverse limits on continua, and in the special case of a continuum X that admits such a sequence of retractions onto arcs, we establish properties that describe the nature of proper subcontinua of X.

Metamorphic testing seeks to verify software in the absence of test oracles. Our application domain is ocean system modeling, where test oracles rarely exist, but where symmetries of the simulated physical systems are known. The input data set is large owing to the requirements of the application domain. This paper presents work in progress for the automated generation of metamorphic test scenarios using machine learning. We extended our previously proposed method [1] to identify metamorphic relations with reduced computational complexity. Initially, we represent metamorphic relations as identity maps. We construct a cost function that minimizes for identifying a metamorphic relation orthogonal to previously found metamorphic relations and penalize for the identity map. A machine learning algorithm is used to identify all possible metamorphic relations minimizing the defined cost function. We propose applying dimensionality reduction techniques to identify attributes in the input which have high variance among the identified metamorphic relations. We apply mutation on these selected attributes to identify distinct metamorphic relations with reduced computational complexity. For experimental evaluation, we subject the two implementations of an ocean-modeling application to the proposed method to present the use of metamorphic relations to test the two implementations of this application., 5 Pages, 1 Figure

In this paper, the stability of the identity map on a compact S-space form is studied. Results related to stability of holomorphic maps on compact domain of S-space form are also discussed.

In this work we give sufficient and necessary conditions for convergence for nonhyperbolic fixed points of dynamical systems of arbitrary dimension whose linearization around zero is the identity function. To achieve this goal, we first rewrite the dynamical system in terms of spherical polar coordinates and by approximation of the radial iteration function we discover a necessary condition depending on a remarkable angular function. Searching for conditions that are sufficient, we discover more angular functions that together with the first one gives a complete set that plays the role of the iteration derivative for unidimensional discrete systems.

A proper (simple) relation algebra is a collection A of binary relations on a set U such that the top element of the algebra is U × U , and A is closed under union, intersection, complementation, composition, converse and contains the identity relation on U. Proper relation algebras can be constructed using U = Z / p Z as a base set using a method due to Comer. The cycle structure of such an algebra must, in general, be determined a posteriori, normally with the aid of a computer. In this paper, we give an improved algorithm for checking the cycle structure that reduces the time complexity from O ( p 2 ) to O ( p ) .

Let be a von Neumann algebra without nonzero central abelian projections on a complex Hilbert space . Let pn (X 1 , X 2 , · · ·, Xn ) be the polynomial defined by n indeterminates X 1, · · ·, Xn and their Jordan multiple ∗-products. In this paper it is shown that a family 𝒟 = {dm } m ∈ℕ of mappings such that , the identity map on satisfies the condition for all U 1 , U 2 , · · ·, Un ∈ and for each m ∈ ℕ if and only if 𝒟 = {dm } m ∈ℕ is an additive ∗-higher derivation. Mathematics Subject Classification (2010): 47B47, 16W25, 46K15.

Computational models of various linguistic term sets whose evaluation scales are the identity functions have been extensively investigated. Among these models, although the virtual linguistic model is the most straightforward and convenient for the calculations, the operations are not closed on the given scales. In this paper, we develop the fuzzy logical algebras-based virtual linguistic model. We propose the generalized linguistic term sets based on general strictly increasing functions with the existing linguistic term sets as special cases, and define some novel operational laws and measures such as similarity measures, distance measures and entropy measures for them. Then we provide some linguistic aggregation operators for the generalized linguistic term sets in multi-attribute decision making. Particularly, when these operators are reduced to the existing ones, it is shown that the proposed operational laws can be considered as a modification of the existing ones in related literature. Based on the proposed operators, a multi-attribute decision-making model is built, and a method for determining the objective–subjective weight vector in generalized linguistic term sets are put forward. Finally, an illustrative example is presented for verifying our models and methods and a comparative analysis with the existing methods is made.

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

We adopt the perspective of similarity equivalence, in gate set tomography called the gauge, to analyze various properties of quantum operations belonging to a semigroup, $\Phi= e^{{\cal L}t}$,and therefore given through the Lindblad operator. We first observe that the non unital part of the channel decouples from the time evolution. Focusing on unital operations we restrict our attention to the single-qubit case, showing that the semigroup embedded inside the tetrahedron of Pauli channels is bounded by the surface composed of product probability vectors and includes the identity map together with the maximally depolarizing channel. Consequently, every member of the Pauli semigroup is unitarily equivalent to a unistochastic map, describing a coupling with one-qubit environment initially in the maximally mixed state, determined by a unitary matrix of order four., Comment: 10 pages, 2 figures

A number of thinkers suggest that, given certain conditions, it’s possible that any concrete human nature could have been united hypostatically to the second Person of the Trinity. Oliver Crisp argues that a potency to have been possibly hypostatically united to the Logos is an important part of what it means for a human person to be made in the image of God. Against this line of reasoning, and building on an argument in print by Andrew Jaeger, I argue two things: first, that many metaphysics of human persons on offer fail to allow the Logos possibly to unite hypostatically to just any concrete human nature. And this is because, given the necessity of identity (if x = y, necessarily x = y), every metaphysics that deploys an identity relation between a human person and her human nature or some essential component of her human nature fails to allow concrete human natures to be identical to any other person than they in fact are or else to lack some essential component that is identical to any other person than it in fact is. And, in the Incarnation, the Logos does not unite with a person. Second, supposing the preceding line of reasoning goes through, I argue that Oliver Crisp’s thesis about the imago Dei is mistaken, provided that it relies on a metaphysics of human persons that deploys an identity relation between a person and her concrete human nature or else some essential component of her concrete human nature.

LetGbe a group and$$\mathrm{IA}(G)$$IA(G)denote the group of all automorphisms ofG, which induce identity map on the abelianized group$$G_{ab}=G/G'$$Gab=G/G′. Also the group of those$$\mathrm{IA}$$IA-automorphisms which fix the centre element-wise is denoted by$$\mathrm{IA_Z}(G)$$IAZ(G). In the present article, among other results and under some condition we prove that the derived subgroups of finitep-groups, for which$$\mathrm{IA_Z}$$IAZ-automorphisms are the same as central automorphisms, are either cyclic or elementary abelian.

A substantial number of expert and intelligent systems rely on deep learning methods to solve problems in areas such as economics, physics, and medicine. Improving the accuracy of the activation functions used by such methods can directly and positively impact the overall performance and quality of the mentioned systems at no cost whatsoever. In this sense, enhancing the design of such theoretical fundamental blocks is of great significance as it immediately impacts a broad range of current and future real-world deep learning based applications. Therefore, in this paper, we turn our attention to the interworking between the activation functions and the batch normalization, which is practically a mandatory technique to train deep networks currently. We propose the activation function Displaced Rectifier Linear Unit (DReLU) by conjecturing that extending the identity function of ReLU to the third quadrant enhances compatibility with batch normalization. Moreover, we used statistical tests to compare the impact of using distinct activation functions (ReLU, LReLU, PReLU, ELU, and DReLU) on the learning speed and test accuracy performance of standardized VGG and Residual Networks state-of-the-art models. These Convolutional Neural Networks were trained on CIFAR-100 and CIFAR-10, the most commonly used deep learning computer vision datasets. The results showed DReLU speeded up learning in all models and datasets. Besides, statistical significant performance assessments (p

We introduce a function model for the Teichmuller space of a closed hyperbolic Riemann surface. Then we introduce a new metric on the Teichmuller space by using the maximum norm on the function space. We prove that the identity map from the Teichmuller space equipped with the Teichmuller metric to the Teichmuller space equipped with this new metric is uniformly continuous. Moreover, we prove that the inverse of the identity, i.e., the identity map from the Teichmuller space equipped with this new metric to the Teichmuller space equipped with the Teichmuller metric, is continuous (but not uniformly). Therefore, the topology induced by the new metric is the same as the topology induced by the Teichmuller metric on the Teichmuller space. Finally, we give a remark about the pressure metric on the function model and the Weil-Petersson metric on the Teichmuller space.

We introduce a feasible corrector–predictor interior-point algorithm (CP IPA) for solving linear optimization problems which is based on a new search direction. The search directions are obtained by using the algebraic equivalent transformation (AET) of the Newton system which defines the central path. The AET of the Newton system is based on the map that is a difference of the identity function and square root function. We prove global convergence of the method and derive the iteration bound that matches best iteration bounds known for these types of methods. Furthermore, we prove the practical efficiency of the new algorithm by presenting numerical results. This is the first CP IPA which is based on the above mentioned search direction.

Let S be an orientable closed surface with genus at least two. From S × I , for a given orientation-reversing homeomorphism f from S × { 1 } to S × { 0 } , there is an orientable closed 3-manifold M f = S × I / f which is called a mapping torus. It is known that M f admits a canonical Heegaard splitting H 1 ∪ Σ H 2 . By the construction of Namazi [H.Namazi, Topology Appl. 154 (2007), no. 16, 2939-2949], the mapping class group of this Heegaard splitting, denoted by M o d ( Σ ; H 1 , H 2 ) , contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in M o d ( Σ ; H 1 , H 2 ) , whether it is reducible or not. Using the translation length of f in the curve complex, we prove that if f is the identity map or its translation length is at least 8, then each element of M o d ( Σ ; H 1 , H 2 ) is reducible.

A distinguished class of polyhedral cones is considered. For a linear operator $$\mathcal {L}$$ preserving a cone in this class, we prove, under some assumption on the number of edges of the cone, that its spectrum contains exactly one strictly positive real eigenvalue. As an application, we characterize those $$\mathcal {L}$$ that acts as a transposition on the edges of this cone. As other application, we show that if $$\mathcal {L}$$ fixes the edges of an $$(n-1)$$-face of this cone then $$\mathcal {L}$$ must be a positive multiple of the identity map.

A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space E, the infumum of the q such that the identity map $$id_{E}$$ is absolutely $$\left( q,1\right) $$-summing is precisely $$\cot E$$. In the same direction, the Dvoretzky–Rogers Theorem asserts $$id_{E}$$ fails to be absolutely $$\left( p,p\right) $$-summing, for all $$p\ge 1$$. In this note, among other results, we unify both theorems by charactering the parameters q and p for which the identity map is absolutely $$\left( q,p\right) $$-summing. We also provide a result that we call strings of coincidences that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending a classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapien.

In this note we discuss three interconnected problems about dynamics of Hamiltonian or, more generally, just smooth diffeomorphisms. The first two concern the existence and properties of the maps whose iterations approximate the identity map with respect to some norm, e.g., \(C^1\)- or \(C^0\)-norm for general diffeomorphisms and the \(\gamma \)-norm in the Hamiltonian case, and the third problem is the Lagrangian Poincare recurrence conjecture.

We prove an inequality which quantifies the idea that a holomorphic self-map of the disc that perturbs two points is close to the identity function., Comment: 8 pages, to appear in Ann. Acad. Sci. Fenn. Math

Moderate composition as identity holds that there is a generalized identity relation, “being the same portion of reality,” of which composition and numerical identity are distinct species. Composition is a genuine kind of identity; but unlike numerical identity, it fails to satisfy Leibniz’s Law. A composite whole and its parts differ with respect to their numerical properties: the whole is one; the parts (collectively) are many. Moderate composition as identity faces the challenge: how, in the absence of Leibniz’s Law, can one characterize what counts as a genuine kind of identity? This paper explores a promising answer: a genuine kind of identity must satisfy a version of Leibniz’s Law restricted to properties that ascribe qualitative character. Strong composition as identity holds that there is only one identity relation, that it satisfies Leibniz’s Law, and that the parts are identical with the whole that they compose. Strong composition as identity faces the challenge of showing that numerical properties do not provide counterexamples to Leibniz’s Law, and doing so in a way that is compatible with the framework of plural logic that is needed to formulate the theory. The most promising way to do this is to hold that plural logic is fundamental at the level of our representations, but not fundamental at the level of being. At the level of being, portions of reality cannot be characterized as either singular or plural. It turns out that the proposed moderate theory and the proposed strong theory are one and the same. In spite of its many attractions, I reject it. The main issue has to do with whether slice-sensitive emergent properties are possible. I argue that they are, making use both of specific examples and general principles of modal plenitude. I do not claim that my arguments are irresistible. But they cannot be evaded as easily as a related argument against strong composition as identity given by Kris McDaniel. I critically examine McDaniel’s argument to pave the way for my own.

We show that every operator on $L^{p}$, $1

We establish a bijection between rigged configurations and highest weight elements of a tensor product of Kirillov-Reshetikhin crystals for all nonexceptional types. A key idea for the proof is to embed both objects into bigger sets for simply-laced types $A_n^{(1)}$ or $D_n^{(1)}$, whose bijections have already been established. As a consequence we settle the $X=M$ conjecture in full generality for nonexceptional types. Furthermore, the bijection extends to a classical crystal isomorphism and sends the combinatorial $R$-matrix to the identity map on rigged configurations., 30 pages, 2 figures; v2 Referenced Naoi's work in the introduction, clarified some notation; v3 Various additions for more self-containment (e.g., the signature rule) and typos fixed