A function is a relation between a set of inputs and set of outputs where each input is related to exactly one output. An example is given of a function that relates shapes to colors, where each shape maps to one unique color. A function can be written as a set of ordered pairs, where the input comes first and the output second. The domain is the set of inputs, the codomain is the set of possible outputs, and the range is the set of outputs the function actually produces. A function is one-to-one if no two distinct inputs map to the same output, and onto if every element in the codomain is mapped to by at least one input.
Continuity, Removable Discontinuity, Essential Discontinuity. These slides accompany my lectures in differential calculus with BSIE and GenENG students of LPU Batangas
Continuity, Removable Discontinuity, Essential Discontinuity. These slides accompany my lectures in differential calculus with BSIE and GenENG students of LPU Batangas
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
In this lecture 1 we are going to cover :
Equations
Complex Numbers
Quadratic Expressions
Inequalities
Absolute Value Equations & Inequalities
Applications
Affect of Money supply on inflation and GDP.................how our GDP and inflation vary with our Indian economy going up or down...................know thru did prez.........
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
2. In mathematics, a function is a relation
between a set of inputs and a set of
permissible outputs with the property that
each input is related to exactly one output.
For an example of a function, let X be the set
consisting of four shapes: a red triangle, a yellow
rectangle, a green hexagon, and a red square; and
let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape
to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and
each shape is linked to exactly one color. There is no shape that lacks a color and no shape that
has two or more colors. This function will be referred to as the "color-of-the-shape function".
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3. 1. "...each element..." means that every element in X is related
to some element in Y.
(But some elements of Y might not be related to any value,
which is fine.)
2. "...exactly one..." means that a function is single valued. It
will not give back 2 or more results for the same input.
So for example "f(2) = 7 or 9" is not right!
If a relationship does not follow those two rules then
it is not a function ... it would still be a relationship,
just not a function.
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4. In our examples above
1. the set "X" is called the Domain,
2. the set "Y" is called the Codomain, and
3. the set of elements that get pointed to in Y (the actual values produced by the function) is
called the Range.
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5. You can write the input and output of a function as an "ordered pair", such as (4,16).
They are called ordered pairs because the input always comes first, and the output second: (input, output)
So it looks like this: ( x, f(x) )
Example- (4,16) means that the function takes in “4” and gives out “16”.
Set of Ordered Pairs
A function can then be defined as a set of ordered pairs:
For example in above diag. where we had shapes and their particular color, the ordered pairs are-( , ),
( , ),( , ),( , ).
But the function has to be single valued, so we also say
“if it contains ( , ), and ( , ) , then must be equal to ”.
Which is not possible and is just a way of saying that an input of "a" cannot produce two different results.
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6. A function f with domain X and codomain Y is commonly denoted by
f : X Y OR X Y
In this context, the elements of X are called arguments of f. For each argument x, the
corresponding unique y in the codomain is called the function value at x or
the image of x under f. It is written as f(x). One says that f associates y with x or maps x to y.
This is abbreviated by y=f(x)
Moreover in following function i.e. –
"f is a function from (the set of natural numbers) to (the set of integers)“
OR
“domain belongs to natural number and range belongs to integers".
f
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7. If A is any subset of the domain X, then f(A) is the subset of the codomain Y consisting of all
images of elements of A. We say the f(A) is the image of A under f. The image of f is given
by f(X). On the other hand, the inverse image (or preimage, complete inverse image) of a
subset B of the codomain Y under a function f is the subset of the domain X defined by:
For example, the preimage of {4, 9} under the squaring function is the set {−3,−2,2,3}.
Image
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Preimage
8. A function is called one-to-one (or an injective) if f(a) ≠ f(b) for any
two different elements a and b of the domain.
It is called onto (or surjective) if f(X) = Y. That is, it is onto if for every element y in the
codomain there is an x in the domain such that f(x) = y. Finally f is called bijective if it is both
injective and surjective.
The following is example of square function of natural number i.e. f(x)=x2 is both one-to-one
and onto:
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9. The composition of two functions takes the
output of one function as the input of a second one.
That is, the value of x is obtained by first applying f to x to obtain
y = f(x) and then applying g to y to obtain z = g(y). The composition
is only defined when the codomain of f is the domain of g.
Assuming that, the composition in the opposite order need not be
defined. Even if it is, i.e., if the codomain of f is the codomain of g,
it is not in general true that
That is, the order of the composition is important. For example, suppose f(x) = x2
and g(x) = x+1. Then g(f(x)) = x2+1, while f(g(x)) = (x+1)2, which
is x2+2x+1, a different function. Property of Amit Amola.
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9
10. The unique function over a set X that maps each element to itself is called the identity
function for X.
For any set of A, the identity function on A is the function A:A A defined by
A(a)=a for all a A. In terms of ordered pairs,
A={ (a , a) | a A}
The Greek symbol is pronounced “yota”, so that “ A ” is read “yota sub A.”
Under composition, an identity function is "neutral": if f is any function from X to Y, then:
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11. Let f:A B. If there exists a function g:B A such that g o f =IA and f o g=IB , then f is called
an invertible function and g is called the inverse of f. We write, f -1=g.
And clearly
As a simple example, if f converts a temperature in degrees Celsius C to degrees Fahrenheit F,
the function converting degrees Fahrenheit to degrees Celsius would be a suitable f −1.
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12. A real-valued function f is one whose codomain is the set of real numbers or a subset thereof.
If, in addition, the domain is also a subset of the real number, f is a real valued function of a real
variable. The study of such functions is called real analysis.
Example:
Let f:R R be defined by f(x)=2x-3. The domain of
f is R and range f =R since any real number y
can be expressed y =2x-3. Graphically, this line is
represented beside the text. Since range f=R, f is onto.
It is also one-to-one, so being onto and one-to-one, it
is a bijection from R to R. 10 5 0 5 10
10
0
10
20
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13. • differentiable, integrable
• polynomial, rational
• algebraic, transcendental
• odd or even
• convex, monotonic
• holomorphic, meromorphic, entire
• vector-valued
• computable
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14. Property of Amit Amola.
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15. When the ATM card is inserted to the
machine, the program inside is performing a
function to map the number stored in the card
to your current or saving account. This is
basically a one-to-one mapping, i.e., function.
This is probably the most widely and popular
use of function.
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16. Money as a function of time. You never
have more than one amount of money at any
time because you can always add everything
to give one total amount. By understanding
how your money changes over time, you
can plan to spend your money sensibly.
Businessmen find it very useful to plot the
graph of their money over time so that they
can see when they are spending too much.
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17. Temperature as a function of various factors.
Temperature is a very complicated function because it
has so many inputs, including: the time of day, the
season, the amount of clouds in the sky, the strength of
the wind, where you are and many more. But the
important thing is that there is only one temperature
output when you measure it in a specific place. This is
what thermometer deals with and is a very good
example for many to one function.
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18. Location as a function of time. You can never be in two places at the same time. If you were to
plot the graphs of where two people are as a function of time, the place where the lines cross
means that the two people meet each other at that time. This idea is used in logistics, an area of
mathematics that tries to plan where people and items are for businesses.
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19. Property of Amit Amola.
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