A relative tensor calculus is formulated for expressing equations of mathematical physics. A tensor time derivative operator ▽ b a is defined which operates on tensors λia...ib. Equations are written in a rigid, flat, inertial or other coordinate system a, altered to relative tensor notation, and are thereby expressed in general flowing coordinate systems or materials b, c, d, .... Mirror tensor expressions for ▽ b a λic...id and ▽ b a λic...id exist in a relative geometry G (...) if and only if a rigid coordinate system a exists in G, where ▽ b a λic = λ ,0c ic + λkev ckc aic + λ kc ic v b ckc , ▽jcλic = λ ,jc ic + λkcΓ jc kc ie , and v b aic is the velocity of b relative to a with components in c. These operators are convenient in theoretical analyses and can be incorporated into machine programs for the numerical solution of physical problems. (shrink)

The Distributional Compositional Categorical model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that (...) could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. (shrink)

The paper provides an analysis of Giuseppe Vitali’s contributions to differential geometry over the period 1923–1932. In particular, Vitali’s ambitious project of elaborating a generalized differential calculus regarded as an extension of Ricci-Curbastro tensor calculus is discussed in some detail. Special attention is paid to describing the origin of Vitali’s calculus within the context of Ernesto Pascal’s theory of forms and to providing an analysis of the process leading to a fully general notion of covariant derivative. Finally, (...) the reception of Vitali’s theory is discussed in light of Enea Bortolotti and Enrico Bompiani’s subsequent works. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (...) constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

A new gauge theory of gravity on flat spacetime has recently been developed by Lasenby, Doran, and Gull. Einstein’s principles of equivalence and general relativity are replaced by gauge principles asserting, respectively, local rotation and global displacement gauge invariance. A new unitary formulation of Einstein’s tensor illuminates long-standing problems with energy–momentum conservation in general relativity. Geometric calculus provides many simplifications and fresh insights in theoretical formulation and physical applications of the theory.

We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell’s equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.

This paper responds to criticisms levelled by Fodor, Pylyshyn, and McLaughlin against connectionism. Specifically, I will rebut the charge that connectionists cannot account for representational systematicity without implementing a classical architecture. This will be accomplished by drawing on Paul Smolensky's Tensor Product model of representation and on his insights about split-level architectures.

ObjectiveWe aim to investigate the feasibility of using diffusion tensor imaging to evaluate changes in extraocular muscles and lacrimal gland in patients with thyroid-associated ophthalmopathy and to evaluate disease severity.Materials and MethodsA total of 74 participants, including 17 healthy controls, 22 patients with mild TAO, and 35 patients with moderate-severe TAO, underwent 3-Tesla DTI to measure fractional anisotropy and mean diffusivity of the EOMs and LG. Ophthalmological examinations, including visual acuity, exophthalmos, intraocular pressure, and fundoscopy, were performed. FA and MD (...) values were compared among patients with different disease severity. Multiple linear regression was adopted to predict the impact of clinical variables on DTI parameters of orbital soft tissue.ResultsTAO patients’ EOMs and LG showed significantly lower FA values and higher MD compared to HCs’. Moderate-severe TAO patients’ EOMs and LG had dramatically lower FA and higher MD compared with HCs. In addition, only the DTI parameters of the medial rectus were considerably different between mild and moderate-severe TAO patients. Multiple linear regression showed that disease severity had a significant impact on the DTI parameters of orbital soft tissue.ConclusionDTI is a useful tool for detecting microstructural changes in TAO patients’ orbital soft tissue. DTI findings, especially medial rectus DTI parameters, can help to indicate the disease severity in TAO patients. (shrink)

A notion of factorizability for vector-valued measures on a quantum logic L enables us to pass from abstract logics to Hilbert space logics and thereby to construct tensor products. A claim by Kruszynski that, in effect, every orthogonally scattered measure is factorizable is shown to be false. Some criteria for factorizability are found.

The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic (...) in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential. (shrink)

We compute the energy tensor and the energy-momentum tensor for electrodynamics coupled to the current of a charged scalar field and for electrodynamics coupled tothe current of a Dirac spinor field, without using the equations of motion.

We construct all cosmic field tensors which are symmetric rank-two tensor concomitants of a metric and a background metric and which have zero divergence when the background metric satisfies the generalized De Donder condition. The resulting background cosmic field represents an Einstein space-time.

Components of the Ricci and Einstein tensors are expressed in terms of the Gaussian curvatures of elementary two-spaces formed by the orthogonal coordinate planes, and the results are applied to some standard metrics.

In the big data era, sequencing technology has produced a large number of biological sequencing data. Different views of the cancer genome data provide sufficient complementary information to explore genetic activity. The identification of differentially expressed genes from multiview cancer gene data is of great importance in cancer diagnosis and treatment. In this paper, we propose a novel method for identifying differentially expressed genes based on tensor robust principal component analysis, which extends the matrix method to the processing of multiway (...) data. To identify differentially expressed genes, the plan is carried out as follows. First, multiview data containing cancer gene expression data from different sources are prepared. Second, the original tensor is decomposed into a sum of a low-rank tensor and a sparse tensor using TRPCA. Third, the differentially expressed genes are considered to be sparse perturbed signals and then identified based on the sparse tensor. Fourth, the differentially expressed genes are evaluated using Gene Ontology and Gene Cards tools. The validity of the TRPCA method was tested using two sets of multiview data. The experimental results showed that our method is superior to the representative methods in efficiency and accuracy aspects. (shrink)

The notions of ``motion'' and ``conserved quantities'', if applied to extended objects, are already quite non-trivial in Special Relativity. This contribution is meant to remind us on all the relevant mathematical structures and constructions that underlie these concepts, which we will review in some detail. Next to the prerequisites from Special Relativity, like Minkowski space and its automorphism group, this will include the notion of a body in Minkowski space, the momentum map, a characterisation of the habitat of globally conserved (...) quantities associated with Poincare symmetry -- so called Poincare charges --, the frame-dependent decomposition of global angular momentum into Spin and an orbital part, and, last not least, the likewise frame-dependent notion of centre of mass together with a geometric description of the Moeller Radius, of which we also list some typical values. (shrink)

Ongoing electroencephalography signals are recorded as a mixture of stimulus-elicited EEG, spontaneous EEG and noises, which poses a huge challenge to current data analyzing techniques, especially when different groups of participants are expected to have common or highly correlated brain activities and some individual dynamics. In this study, we proposed a data-driven shared and unshared feature extraction framework based on nonnegative and coupled tensor factorization, which aims to conduct group-level analysis for the EEG signals from major depression disorder patients and (...) healthy controls when freely listening to music. Constrained tensor factorization not only preserves the multilinear structure of the data, but also considers the common and individual components between the data. The proposed framework, combined with music information retrieval, correlation analysis, and hierarchical clustering, facilitated the simultaneous extraction of shared and unshared spatio-temporal-spectral feature patterns between/in MDD and HC groups. Finally, we obtained two shared feature patterns between MDD and HC groups, and obtained totally three individual feature patterns from HC and MDD groups. The results showed that the MDD and HC groups triggered similar brain dynamics when listening to music, but at the same time, MDD patients also brought some changes in brain oscillatory network characteristics along with music perception. These changes may provide some basis for the clinical diagnosis and the treatment of MDD patients. (shrink)

PurposeTo explore the changes of cerebral blood flow and fractional anisotropy in stroke patients with motor dysfunction after repetitive transcranial magnetic stimulation treatment, and to better understand the role of rTMS on motor rehabilitation of subcortical stroke patients from the perfusion and structural level.Materials and MethodsIn total, 23 first-episode acute ischemic stroke patients and sixteen healthy controls were included. The patients were divided into the rTMS and sham group. The rehabilitation assessments and examination of perfusion and structural MRI were performed (...) before and after rTMS therapy for each patient. Voxel-based analysis was used to detect the difference in CBF and FA among all three groups. The Pearson correlation analysis was conducted to evaluate the relationship between the CBF/FA value and the motor scales.ResultsAfter rTMS, significantly increased CBF was found in the ipsilesional supplementary motor area, postcentral gyrus, precentral gyrus, pons, medulla oblongata, contralesional midbrain, superior cerebellar peduncle, and middle cerebellar peduncle compared to that during the prestimulation and in the sham group, these fasciculi comprise the cortex-pontine-cerebellum-cortex loop. Besides, altered CBF in the ipsilesional precentral gyrus, postcentral gyrus, and pons was positively associated with the improved Fugl-Meyer assessment scores. Significantly decreased FA was found in the contralesional precentral gyrus, increased FA was found in the ipsilesional postcentral gyrus, precentral gyrus, contralesional supplementary motor area, and bilateral cerebellum, these fasciculi comprise the corticospinal tract. The change of FMA score was positively correlated with altered FA value in the ipsilesional postcentral gyrus and negatively correlated with altered FA value in the contralesional precentral gyrus.ConclusionOur results suggested that rTMS could facilitate the motor recovery of stroke patients. High frequency could promote the improvement of functional activity of ipsilesional CPC loop and the recovery of the microstructure of CST. (shrink)

In [5] we study Nonassociative Lambek Calculus augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical Non-Associative Lambek Calculus. Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a one-sided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.

The λY calculus is the simply typed λ calculus augmented with the fixed point operators. We show three results about λY: the word problem is undecidable, weak normalisability is decidable, and higher type fixed point operators are not definable from fixed point operators at smaller types.