7,792 research outputs found
Derivatives of tangent function and tangent numbers
In the paper, by induction, the Fa\`a di Bruno formula, and some techniques
in the theory of complex functions, the author finds explicit formulas for
higher order derivatives of the tangent and cotangent functions as well as
powers of the sine and cosine functions, obtains explicit formulas for two Bell
polynomials of the second kind for successive derivatives of sine and cosine
functions, presents curious identities for the sine function, discovers
explicit formulas and recurrence relations for the tangent numbers, the
Bernoulli numbers, the Genocchi numbers, special values of the Euler
polynomials at zero, and special values of the Riemann zeta function at even
numbers, and comments on five different forms of higher order derivatives for
the tangent function and on derivative polynomials of the tangent, cotangent,
secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.Comment: 17 page
Joint Intermodal and Intramodal Label Transfers for Extremely Rare or Unseen Classes
In this paper, we present a label transfer model from texts to images for
image classification tasks. The problem of image classification is often much
more challenging than text classification. On one hand, labeled text data is
more widely available than the labeled images for classification tasks. On the
other hand, text data tends to have natural semantic interpretability, and they
are often more directly related to class labels. On the contrary, the image
features are not directly related to concepts inherent in class labels. One of
our goals in this paper is to develop a model for revealing the functional
relationships between text and image features as to directly transfer
intermodal and intramodal labels to annotate the images. This is implemented by
learning a transfer function as a bridge to propagate the labels between two
multimodal spaces. However, the intermodal label transfers could be undermined
by blindly transferring the labels of noisy texts to annotate images. To
mitigate this problem, we present an intramodal label transfer process, which
complements the intermodal label transfer by transferring the image labels
instead when relevant text is absent from the source corpus. In addition, we
generalize the inter-modal label transfer to zero-shot learning scenario where
there are only text examples available to label unseen classes of images
without any positive image examples. We evaluate our algorithm on an image
classification task and show the effectiveness with respect to the other
compared algorithms.Comment: The paper has been accepted by IEEE Transactions on Pattern Analysis
and Machine Intelligence. It will apear in a future issu
Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals
We demonstrate experimentally that the interactions between a pair of
nonlocal spatial optical solitons in a nematic liquid crystal (NLC) can be
controlled by the degree of nonlocality. For a given beam width, the degree of
nonlocality can be modulated by varying the pretilt angle of NLC molecules via
the change of the bias. When the pretilt angle is smaller than pi/4, the
nonlocality is strong enough to guarantee the independence of the interactions
on the phase difference of the solitons. As the pretilt angle increases, the
degree of nonlocality decreases. When the degree is below its critical value,
the two solitons behavior in the way like their local counterpart: the two
in-phase solitons attract and the two out-of-phase solitons repulse.Comment: 3 pages, 4 figure
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